The inverse Laplace transform of s² + 2s + 10 / (3s + 2) using the First Translation Theorem is (1/3) * [tex]e^{-2/3t[/tex] * δ(t) + (2/3) * [tex]e^{-2/3t}[/tex] + (10/3) * [tex]e^{-2/3t}[/tex].
To evaluate L⁻¹{s² + 2s + 10 / (3s + 2)}, we can use the First Translation Theorem along with the known Laplace transforms for certain functions.
First, let's rewrite the expression in terms of a shifted variable:
L⁻¹{s² + 2s + 10 / (3s + 2)} = L⁻¹{(s² + 2s + 10) / (3(s + 2/3))}
According to the First Translation Theorem, for a function f(t) with Laplace transform F(s), we have:
L⁻¹{F(s - a)} = e^(at) * L⁻¹{F(s)}.
Now, let's apply the First Translation Theorem to the terms in the expression:
L⁻¹{s² / (3(s + 2/3))} = [tex]e^{-2/3t}[/tex] * L⁻¹{(s²) / (3s)} = [tex]e^{-2/3t}[/tex] * (1/3) * L⁻¹{s} = [tex]e^{-2/3t}[/tex] * (1/3) * δ(t).
Here, δ(t) represents the Dirac delta function.
L⁻¹{2s / (3(s + 2/3))} = [tex]e^{-2/3t}[/tex] * L⁻¹{(2s) / (3s)} = [tex]e^{-2/3t}[/tex] * (2/3) * L⁻¹{1} = [tex]e^{-2/3t}[/tex] * (2/3) * 1 = (2/3) * [tex]e^{-2/3t}[/tex].
L⁻¹{10 / (3(s + 2/3))} = [tex]e^{-2/3t}[/tex] * L⁻¹{10 / (3s)} = [tex]e^{-2/3t}[/tex] * (10/3) * L⁻¹{1} = [tex]e^{-2/3t}[/tex] * (10/3) * 1 = (10/3) * [tex]e^{-2/3t}[/tex].
Finally, combining the results:
L⁻¹{s² + 2s + 10 / (3s + 2)} = (1/3) * [tex]e^{-2/3t}[/tex] * δ(t) + (2/3) * [tex]e^{-2/3t}[/tex] + (10/3) * [tex]e^{-2/3t}[/tex].
Therefore, using the First Translation Theorem, the inverse Laplace transform of s² + 2s + 10 / (3s + 2) is (1/3) * [tex]e^{-2/3t}[/tex] * δ(t) + (2/3) * [tex]e^{-2/3t}[/tex] + (10/3) * [tex]e^{-2/3t}[/tex].
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Use the Integral Test to determine whether the series converges. Show all work to justify your answer. 11) ∑ n=1
[infinity]
n 2
+5
3n
The given series is divergent.
Given series is as follows,∑n=1∞[n2+53n]
We need to find whether the given series is convergent or divergent.
Integral test: The integral test, also known as the Cauchy integral test, is a test used to decide the convergence or divergence of a series.
It is commonly used to check the convergence of p-series that contain non-integer powers. It is also applicable to series containing alternating terms.
According to this test, if a function f (x) is continuous, non-negative, and monotonically decreasing over the interval [1, ∞), then the infinite series Σ f (n) converges if and only if the improper integral ∫1∞f(x)dx converges.
Let us evaluate the integral of the given series for n = 1 and n = ∞∫1∞(x2+5)/(3x)dx
Let us take out 1/3 as a common factor∫1∞x−1(dx)+5/3∫1∞x−2(dx)
Now, we need to integrate both the terms
∫1∞x−1(dx)= ln x |1∞
=- ln 1 + ln ∞
= ∞∫1∞x−2(dx)
= −x−1 |1∞
=0 − (−1)
= 1
Therefore, we have∫1∞(x2+5)/(3x)dx= 1/3(∞ + 5)= ∞
The integral value comes out to be infinity which means that the given series diverges.
Therefore, the given series is divergent.
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2. Find the area under the graph of y=2-x-x² over the interval [-2, 1].
The area under the graph of y=2−x−x² over the interval [-2, 1] can be calculated using definite integral. Definite integral is a fundamental concept in calculus that is used to calculate the signed area of the region between the curve and the x-axis over an interval.
The given function is y = 2 - x - x²We are supposed to calculate the area under the graph of this function over the interval [-2, 1]. The graph of the function is shown below: The area under the curve of the function can be divided into two parts: one above the x-axis and the other below the x-axis. Area above the x-axis:
This area can be calculated using definite integral as follows: Area below the x-axis:This area is negative and can be calculated as follows: Hence, the area under the graph of y=2−x−x² over the interval .
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If θ is the angle between ⟨2,1,−2⟩ and ⟨3,−4,0⟩, then cosθ= ⟨2,1,−2⟩ ve ⟨3,−4,0⟩θ ise cosθ= A. - 3/5 B. - 2/3 C. - 3/2 D. - 0 E. - 5/15
The value of cos θ is: cosθ=⟨u, v⟩ / ||u|| ||v|| = 2 / (3 * 5) = 2/15
The correct option is E. cos θ = -5/15.
To find the value of cos θ, we use the formula:
cosθ=⟨u, v⟩ / ||u|| ||v||
where u and v are two vectors.
Given the vectors are ⟨2,1,−2⟩ and ⟨3,−4,0⟩, the dot product is:
⟨2,1,−2⟩ · ⟨3,−4,0⟩ = (2 * 3) + (1 * -4) + (-2 * 0)
= 6 - 4
= 2
Now, calculating the magnitudes, we get:
||⟨2,1,−2⟩|| = √(2² + 1² + (-2)²)
= √9
= 3
and,
||⟨3,−4,0⟩|| = √(3² + (-4)² + 0²)
= √25
= 5
Therefore, the value of cos θ is:cosθ=⟨u, v⟩ / ||u|| ||v|| = 2 / (3 * 5) = 2/15
The correct option is E. cos θ = -5/15.
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Find the Laplace transform of (a) f(t) = e2t cosh² t (b) f(t) = tsin 6t (c) t³8 (t-1) Given the piecewise continuous function f(t) = 1, 0, e-4t, 0 < x < 2, 2 < x < 4, t> 4. (a) Express the above function in terms of unit step functions. (b) Hence, find the Laplace transform of f(t)
(a) The Laplace transform of f(t) = e2t cosh² t is { s - 2 } / { ( s - 2 )² - 4 }
(b) the Laplace transform of f(t) = tsin 6t is { 6 } / { ( s² + 36 )² }
(c) the Laplace transform of f(t) = t³8 (t-1) is { 120 } / { s⁶ } - { 24 } / { s⁵ }
(a) f(t) in terms of unit step functions is; f(t) = u(t) - u(t - 2) + e-4tu(t - 4)
(b) the Laplace transform of f(t) in terms of unit step functions is { s + 2e-2s - 1 } / { s( s + 4 ) }.
(a) f(t) = e2t cosh² t
To find the Laplace transform of f(t) = e2t cosh² t use the following formulas as shown:
=> [tex]L ( e^{at} cosh(bt) ) = { s - a } / { ( s - a )^2 - b^2 }[/tex]
=>[tex]L ( cosh^2(at) ) = { s } / { ( s - a ) ( s + a ) }[/tex]
As cosh is an even function, the transform is given as;
[tex]L ( e^{at} cosh(bt) ) = { s - a } / { ( s - a )^2 - b^2 }[/tex]
On substituting the values of a and b,
L { e2t cosh² t } = { s - 2 } / { ( s - 2 )² - 4 }
(b) f(t) = tsin 6t
The Laplace transform of f(t) = tsin 6t is given as;
=> L { sin ( at ) } = { a } / { s² + a² }
=> L { t } = { 1 } / { s² }
On substituting the values of a and b,
L { tsin 6t } = { 6 } / { ( s² + 36 )² }
Therefore, the Laplace transform of f(t) = tsin 6t is { 6 } / { ( s² + 36 )² }
(c) t³8 (t-1)The Laplace transform of f(t) = t³8 (t-1) is given as:
=> L { tⁿ f(t) } = { (-1) }ⁿ dⁿ F(s) / dsⁿ
Using this formula, obtain the transform as follows:
L { t⁴ ( t - 1 ) } = L { t⁵ - t⁴ }=> L { t⁵ - t⁴ } = { 5! } / { s⁶ } - { 4! } / { s⁵ }
On simplifying the expression,
L { t³8 ( t - 1 ) } = { 120 } / { s⁶ } - { 24 } / { s⁵ }
Therefore, the Laplace transform of f(t) = t³8 (t-1) is { 120 } / { s⁶ } - { 24 } / { s⁵ }
(a) Express the above function in terms of unit step functions. Given the piecewise continuous function f(t) = 1, 0, e-4t, 0 < x < 2, 2 < x < 4, t> 4. In terms of unit step functions, the given function can be expressed as follows:
f(t) = 1{ t > 0 } - 1{ t > 2 } + e-4t{ t > 4 }
Therefore, f(t) in terms of unit step functions is; f(t) = u(t) - u(t - 2) + e-4tu(t - 4)
(b) Hence, find the Laplace transform of f(t)Using the linearity property of Laplace transforms,
L { f(t) } = L { u(t) } - L { u(t - 2) } + L { e-4tu(t - 4) }
The Laplace transform of unit step function is given by;
L { u(t - a) } = e-as / s
On substituting the values of a and s,
L { u(t - 2) } = e-2s / s
Similarly, the Laplace transform of exponential function is given as;
L { e-at } = { 1 } / { s + a }
On substituting the values of a and s,
L { e-4t } = { 1 } / { s + 4 }
Therefore, the Laplace transform of f(t) in terms of unit step functions is given as:
L { f(t) } = 1/s - e-2s/s + { 1 } / { ( s + 4 ) }
On simplifying,
L { f(t) } = { s + 2e-2s - 1 } / { s( s + 4 ) }
Therefore, the Laplace transform of f(t) is { s + 2e-2s - 1 } / { s( s + 4 ) }.
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How long will it take for \( \$ 2,300 \) to double if it is invested at 10 . decimal placos: It will take years to double. How long will it take if the interest is compounded continuously" Compounded
The amount of time it will take for $2,300 to double if it is invested at 10 percent depends on whether the interest is compounded continuously or not. If the interest is compounded continuously, it will take approximately 6.93 years to double the investment.
The formula for continuous compound interest is:A = P*e^(rt)where
A = final amount
P = initial principal
r = annual interest rate
t = time
The initial principal is $2,300 and the final amount is $4,600 (double of the initial principal).
The annual interest rate is 10% (0.10 in decimal form).
The time is unknown, and it is represented by t.$4,600
= $2,300*e^(0.10t)
Divide both sides by $2,300:e^(0.10t) = 2.00
Take the natural log of both sides:ln(e^(0.10t))
= ln(2.00)0.10t
= 0.6931
Solve for t:
t = 0.6931 / 0.10t
≈ 6.93
Therefore, if the interest is compounded continuously, it will take approximately 6.93 years to double the investment.
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answer the question below.
Answer:
32°----------------------
According to the diagram we see that:
ΔABC is a right triangle since ∠B is a right angle because all interior angles of a rectangle are right angles.Therefore, ∠BAC and ∠BCA are complementary. We have the measure of one of the two, hence the other one is:
m∠BAC = 90° - m∠BCAm∠BAC = 90° - 58°m∠BAC = 32°If The Series ∑Cnxn Converges At X=4, Then The Series Check All That Apply. Cn(−3)N Also Converges. Cn(5)N Converges Cn Diverges
The statement "Cn(-3)^n also converges" is true, while the statements "Cn(5)^n converges" and "Cn diverges" cannot be determined solely based on the convergence of the series at x = 4.
If the series ∑Cnxn converges at x = 4, we can draw the following conclusions:
1. Cn(-3)^n also converges: If the original series converges at x = 4, it implies that the series converges within the interval of convergence, which includes x = -3. Therefore, Cn(-3)^n also converges.
2. Cn(5)^n converges: Since the series converges at x = 4, it does not necessarily mean that the series will converge at other values of x, such as x = 5. The convergence of the series at a specific value of x does not guarantee convergence at other values.
3. Cn diverges: The convergence of the series at x = 4 does not provide any information about the individual terms Cn. The series may converge due to the specific arrangement and behavior of the terms Cn*x^n, but it does not imply anything about the convergence or divergence of the sequence {Cn} itself.
To summarize, the statement "Cn(-3)^n also converges" is true, while the statements "Cn(5)^n converges" and "Cn diverges" cannot be determined solely based on the convergence of the series at x = 4.
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Which of the following choices lists the sides from shortest to longest?
CB, AB, AC
CB, AC, AB
AC, AB, CB
None of these choices are correct.
Step-by-step explanation:
Using the theorem, the side opposite of the largest angles is the largest side, therefore the side opposite of the smallest angle is the smallest side.
So the answer is Option D
Answer:
none of these choices are correct
Step-by-step explanation:
the larger the angle the longer the opposing side.
the smallest angle is B (58°). the opposing side AC is therefore the shortest.
then A (59°) and CB.
and finally C (63°) and AB.
so, the correct sequence would be
AC, CB, AB
this is none of the provided answer options.
so, the 4th option
none of these choices are correct
is correct.
Identify each of the following from the figure where ABCD is a
parallelogram.
a) A pair of supplementary angles:
____________________________
b) A pair of congruent line segments:
____________________
a) A pair of supplementary angles:Supplementary angles are angles whose sum is 180°. We can see that angle A and angle D are adjacent angles in a parallelogram ABCD as shown below:Supplementary angles.
Therefore, the pair of supplementary angles in the figure is angle A and angle D.
b) A pair of congruent line segments:Congruent line segments are line segments that have the same length.
We can see that line segment AB and line segment DC are opposite sides of the parallelogram ABCD.
As we know, opposite sides of a parallelogram are congruent.
Therefore, the pair of congruent line segments in the figure is line segment AB and line segment DC.
Hence, we have identified the pair of supplementary angles and a pair of congruent line segments in the given figure.
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In examining the scoring of students for business statistics, a sample of 16 students revealed that they score, on average, 65 percent. Based on previous semesters, the population standard deviation is thought to be 53 percent. Assuming that the scores are normally distributed, find a 95% confidence interval for the average score. A. [39.03, 90.97] B. [34.18, 95.82] C. [30.87, 99.13] D. [18.12, 100.00]
The 95% confidence interval for the average score is [34.18, 95.82]. Therefore, the correct option is (B) [34.18, 95.82].
Given that: Sample of 16 students revealed that they score, on average, 65 percent. And previous semesters, the population standard deviation is thought to be 53 percent.
To find: The 95% confidence interval for the average score.
Here, the sample size is 16, which is less than 30. Hence, we can use the t-distribution to estimate the population parameter.
The formula for the confidence interval is given by:(x‾ - t_(α/2) * (s/√n), x‾ + t_(α/2) * (s/√n)).
Here, n = 16 (sample size)x‾ = 65 (sample mean)σ = 53 (population standard deviation).
We need to find the t-value corresponding to 95% confidence level and 15 degrees of freedom.α = 1 - 0.95 = 0.05 (as 95% confidence interval is given)So, α/2 = 0.025 (two-tailed test)At 15 degrees of freedom, the t-value for 0.025 is 2.131.
Confidence interval = (x‾ - t_(α/2) * (s/√n), x‾ + t_(α/2) * (s/√n))(65 - 2.131 * (53/√16), 65 + 2.131 * (53/√16))= (34.18, 95.82)Therefore, the correct option is (B) [34.18, 95.82].
To find the 95% confidence interval for the average score we have used the formula:(x‾ - t_(α/2) * (s/√n), x‾ + t_(α/2) * (s/√n)).
Here, n = 16 (sample size)x‾ = 65 (sample mean)σ = 53 (population standard deviation)We need to find the t-value corresponding to 95% confidence level and 15 degrees of freedom.α = 1 - 0.95 = 0.05 (as 95% confidence interval is given)So, α/2 = 0.025 (two-tailed test).
At 15 degrees of freedom, the t-value for 0.025 is 2.131.Confidence interval = (x‾ - t_(α/2) * (s/√n), x‾ + t_(α/2) * (s/√n))(65 - 2.131 * (53/√16), 65 + 2.131 * (53/√16))= (34.18, 95.82)
The 95% confidence interval for the average score is [34.18, 95.82]. Therefore, the correct option is (B) [34.18, 95.82].Note: The general formula for the confidence interval in case of a small sample (n < 30) is(x‾ - t_(α/2) * (s/√n - 1), x‾ + t_(α/2) * (s/√n - 1))Where n is the sample size, x‾ is the sample mean, s is the sample standard deviation, α is the significance level, and t_(α/2) is the critical value of t at α/2 with n - 1 degrees of freedom.
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This velocity time graph shows the motion of a speed boat........ :)
The total distance travelled by the speedboat in 1 d.p is 137.5m
What is velocity-time graph?A velocity-time graph shows the changing velocity of the sprinter or of any other moving person or object.
The rate of change of displacement with time is velocity.
In a velocity-time graph, the area of the shape of the graph is the total distance travelled.
The shape of this graph is Triangle.
Therefore the area of a triangle is expressed as;
A = 1/2bh
where b is the base and h is the height.
base = 11 and height = 25
A = 1/2 × 11 × 25
A = 275/2
A = 137.5m
Therefore the distance covered by the speedboat is 137.5m
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How many months (nearest whole month) do you need to triple your money if you invest $5000 now into a bank account earning an annual interest rate of 9.6% compounded monthly if you want to triple your investment? Use the formula A=P(1+r/n)^(nt).
The value is t_months = ln(3) / (12 ln(1 + 0.096/12)) * 12. this expression gives us the number of months it takes to triple the investment.
To determine how many months it takes to triple your money, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Principal amount (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case, we have:
P = $5000
r = 9.6% = 0.096 (as a decimal)
n = 12 (compounded monthly)
We want to find t, the number of years (or months) it takes to triple the investment. The final amount, A, will be 3 times the principal amount:
A = 3P
Substituting the given values into the formula:
3P = P(1 + r/n)^(nt)
Now we can solve for t. Dividing both sides of the equation by P:
3 = (1 + r/n)^(nt)
Taking the natural logarithm of both sides:
ln(3) = nt ln(1 + r/n)
Now, solving for t:
t = ln(3) / (n ln(1 + r/n))
Substituting the given values:
t = ln(3) / (12 ln(1 + 0.096/12))
Calculating this expression will give us the number of years. To convert it to months, we multiply by 12:
t_months = t * 12
Now, we can calculate the value:
t_months = ln(3) / (12 ln(1 + 0.096/12)) * 12
Evaluating this expression gives us the number of months it takes to triple the investment.
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PLEASE HELP I DONT KNOW MAN ......................
The summation expression [tex]\sum\limits^{\infty}_{n=1} {\frac34(\frac43)^{n-1}[/tex] when evaluated is -2 1/4
How to evaluate the summation expressionFrom the question, we have the following parameters that can be used in our computation:
[tex]\sum\limits^{\infty}_{n=1} {\frac34(\frac43)^{n-1}[/tex]
From the above summation expression, we have the following
First term, a = 3/4Common ratio, r = 4/3The sum to infinity of the sequence is then represented as
Sum = a/(1 - r)
So, we have
[tex]\sum\limits^{\infty}_{n=1} {\frac34(\frac43)^{n-1} = \frac{3/4}{1 - 4/3}[/tex]
Evaluate the difference
[tex]\sum\limits^{\infty}_{n=1} {\frac34(\frac43)^{n-1} = \frac{3/4}{-1/3}[/tex]
Evaluate the quotient
[tex]\sum\limits^{\infty}_{n=1} {\frac34(\frac43)^{n-1} = -2 \frac14[/tex]
Hence, the solution is [tex]\sum\limits^{\infty}_{n=1} {\frac34(\frac43)^{n-1} = -2 \frac14[/tex]
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-6 4 6 4- 2+ 2 4- 6 +3x+1/2y=-3 X What value of b will cause the system to have an infinite number of solutions? y = 6x-b -3x+²y=-3 b = 2 4 6 8
Answer:
Step-by-step explanation:
To determine the value of b that will cause the system to have an infinite number of solutions, we need to analyze the given system of equations:
Equation 1: 4x - 6 + 3x + (1/2)y = -3
Equation 2: y = 6x - b
Equation 3: -3x + 2y = -3
To have an infinite number of solutions, the two equations must represent the same line. This occurs when the coefficients of the x and y terms and the constant terms are proportional.
Let's compare Equation 2 and Equation 3 to find the value of b:
From Equation 2: y = 6x - b
From Equation 3: -3x + 2y = -3
To make the coefficients of x and y proportional, we need to ensure that the ratios of the coefficients are equal. In this case, we compare the coefficients of x:
6 from Equation 2 and -3 from Equation 3
For these coefficients to be proportional, we can multiply the coefficient from Equation 2 by -2:
-2 * 6 = -12
Now let's compare the coefficients of y:
1 from Equation 2 and 2 from Equation 3
For these coefficients to be proportional, we can multiply the coefficient from Equation 2 by 2:
2 * 1 = 2
Comparing the constant terms:
There is a constant term of 0 in Equation 2, while the constant term in Equation 3 is -3.
To make these constant terms proportional, we need to multiply the constant term in Equation 2 by 0:
0 * 0 = 0
Now we have the following comparison:
-12 (coefficient of x) : -3 (coefficient of x) : 2 (coefficient of y) : 2 (coefficient of y) : 0 (constant term) : -3 (constant term)
The ratios of all these terms are equal, which means the two equations represent the same line. Therefore, the value of b that will cause the system to have an infinite number of solutions is b = 0.
The value of 'b' that would cause the system y = 6x - b and -3x + 2y = -3 to have an infinite number of solutions is 7.5. This is achieved by making the 'b' of the first equation equal to 1.5 of the second equation, thus making the two equations equivalent.
Explanation:This question involves the topic of algebra, specifically systems of equations. To determine what value of 'b' will make the system of equations have an infinite number of solutions, we must set the two equations equal to each other. In other words, finding a 'b' value that makes the equation y = 6x - b identical to -3x + 2y = -3.
Firstly
, we need to rearrange the second equation to fit the format of y = mx + b. This gives us y = 1.5x + 1.5.
Secondly
, we must look for a 'b' value in the first equation that makes it identical or equivalent to the second equation; in this case, that would be
b = 7.5
, as it makes the two equations identical, thus resulting in an infinite number of solutions.
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Need help ASAP! PLEASE
Answer:
Step-by-step explanation:
3
Answer:
6
Step-by-step explanation:
triangle area
1/2bh
1/2 * 4 * 6
= 12
same area as parallelogram
parallelogram area
= bh
= 2 * ___
to get 12
= 2 * 6
= 12
so 6
Find the exact value of the expression whenever it is defined. (If an answer is undefined, enter UNDEFINED.) (a) arcsin(sin13π/12) (b) arccos(cos 8π/5) (c) arctan(tan 11π/9)
a. the exact value of arcsin(sin(13π/12)) is **π/12**. b. the value of arccos(cos(8π/5)) is **UNDEFINED**. c. the exact value of arctan(tan(11π/9)) is **5π/9**.
(a) To find the exact value of arcsin(sin(13π/12)), we need to determine the angle whose sine is equal to sin(13π/12). However, it's important to note that the range of the arcsin function is [-π/2, π/2].
The reference angle for 13π/12 is π - (13π/12) = π/12, which lies in the range of the arcsin function. Furthermore, the sine function is positive in both the first and second quadrants, so the angle will be positive.
Therefore, the exact value of arcsin(sin(13π/12)) is **π/12**.
(b) For arccos(cos(8π/5)), we need to find the angle whose cosine is equal to cos(8π/5). The range of the arccos function is [0, π].
The reference angle for 8π/5 is 8π/5 - 2π = -2π/5, which is negative and not within the range of the arccos function. Hence, the expression is undefined.
Therefore, the value of arccos(cos(8π/5)) is **UNDEFINED**.
(c) In the case of arctan(tan(11π/9)), we are looking for the angle whose tangent is equal to tan(11π/9). The range of the arctan function is (-π/2, π/2).
The reference angle for 11π/9 is 11π/9 - 2π = 5π/9, which is within the range of the arctan function. Therefore, the exact value of arctan(tan(11π/9)) is **5π/9**.
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Find y +
and y ′
SCALCET9 3. XP. 4.055 In Gapeer 1 1
we medeled the world pooulation from 1900 to 2010 with the evenentiai function x(n)=(143.53)−(1.01365) t
to tho decimal places.) 1920 1930 2000
The rate of some function tells the increment in output values per unit increment in input.
The rate of increase of world population in 1920 was 17.02
In 1955 it was 42.64, and in 2000 it was 79.53 approx.
Here, we have,
to find the rate of an exponential function:
Suppose that the exponential function is given as
f(x) = a × bˣ
Then its rate (first derivative) (assuming differentiable) with respect to x is given as:
f'(x) = abˣ ln(b)
Since the given function for population measure as a function of time is
x(n)=(143.53)×(1.01365)ⁿ
Its rate is given as
x'(n) = 1.99×(1.01365)ⁿ
Since time was starting from t = 0 (year 1900), so at 1920, the value of t is 20.
Thus, rate of increase in world's population at year 1920 was
x'(20) = 26.25
Similarly,
x'(55) = 42.64
x'(100) = 79.53
Thus,
The rate of increase of world population in 1920 was 17.02
In 1955 it was 42.64, and in 2000 it was 79.53 approx.
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Suppose that: sin(A)/a=sin(B)/b Solve the equation above for A, given that B=39∘,a=17, and b=12. MathPad symbol drawer or type deg. For example, sin(30deg). A=
The answer in symbolic form, we will keep it as an exact expression \(A = \sin^{-1}\left(\frac{{17 \times \sin(39°)}}{12}\right)\)
To solve the equation \(\frac{{\sin(A)}}{a} = \frac{{\sin(B)}}{b}\) for A, given that B = 39°, a = 17, and b = 12, we can substitute the known values and solve for A.
\(\frac{{\sin(A)}}{17} = \frac{{\sin(39°)}}{12}\)
To isolate \(\sin(A\)), we can cross-multiply:
\(\sin(A) \times 12 = 17 \times \sin(39°)\)
Now, divide both sides by 12:
\(\sin(A) = \frac{{17 \times \sin(39°)}}{12}\)
To solve for A, we can take the inverse sine (or arcsine) of both sides:
\(A = \sin^{-1}\left(\frac{{17 \times \sin(39°)}}{12}\right)\)
Using a calculator, we can evaluate the right-hand side to find the approximate value of A. However, since you specified in the question that you want the answer in symbolic form, we will keep it as an exact expression:
\(A = \sin^{-1}\left(\frac{{17 \times \sin(39°)}}{12}\right)\)
Please note that the result is an exact value, and it can also be expressed in degrees or radians depending on the mode of your calculator.
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radius of 4000miles while flying 10miles above the surface. something unusal on the horizon infront of her, how far away is she
The pilot can see up to a distance of approximately 144 miles from her position. However, it is important to note that this distance may vary based on atmospheric conditions and other factors.
Assuming the earth is a perfect sphere, the radius of the earth is approximately 4000 miles.
If the pilot is flying 10 miles above the surface, then her altitude is 4010 miles.
Now, let's assume that she sees something unusual on the horizon in front of her.
The horizon is the line where the sky appears to meet the earth's surface when you look out into the distance.
It is the farthest distance that you can see because the earth's curvature blocks anything beyond that point.
The distance of the horizon from an observer is directly proportional to the height of the observer above the surface of the earth.
The following formula can be used to calculate the distance of the horizon from an observer:
Distance to horizon = sqrt(2Rh + h^2)where R is the radius of the earth and h is the height of the observer above the surface of the earth.
If we substitute the given values into the formula, we can find the distance of the horizon from the pilot: Distance to horizon = sqrt(2 * 4000 + 10^2)≈ 144.07 miles
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A company issued 50 bonds of P1,000 face value each, redeemable at par at the ends of 15 years to accumulate the funds required for redemption, the firm restablished a sinking fund consisting of annual deposits, the interest rate being 4%. Find the following. Redemption value *Letters only Annual deposits a. 2,376 b. 2,460 c. 2,497 d. 2,566 e. 2,675 The principal in the fund at end of 12th year
The correct Redemption value: P50,000Annual deposits: Approximately P2,670.21
Principal in the fund at the end of the 12th year: Approximately P4,278.24
To find the redemption value of the bonds, we need to calculate the future value of the sinking fund.
Given that the face value of each bond is P1,000 and there are 50 bonds, the total redemption value is 50 * P1,000 = P50,000.
To accumulate the funds required for redemption, annual deposits are made into a sinking fund. The interest rate is 4% and the period is 15 years.
To calculate the annual deposits, we can use the formula for the future value of an ordinary annuity:
[tex]FV = R * ((1 + r)^n - 1) / r[/tex]
where FV is the future value, R is the annual deposit, r is the interest rate per period, and n is the number of periods.
Using the given values, we can solve for the annual deposits:
50,000 = [tex]R * ((1 + 0.04)^15 - 1) / 0.04[/tex]
Simplifying the equation:
50,000 = R * (1.749006 - 1) / 0.04
50,000 = R * 0.749006 / 0.04
50,000 = R * 18.72515
Dividing both sides by 18.72515:
R = 50,000 / 18.72515
R ≈ 2,670.21
Therefore, the annual deposits are approximately P2,670.21.
To find the principal in the fund at the end of the 12th year, we can calculate the future value of the sinking fund after 12 years.
Using the formula for the future value of a sinking fund:
[tex]FV = P * (1 + r)^n[/tex]
where P is the annual deposit, r is the interest rate per period, and n is the number of periods.
Substituting the values:
FV = 2,670.21 * (1 + 0.04)^12
FV ≈ 2,670.21 * 1.601031
FV ≈ 4,278.24
Therefore, the principal in the fund at the end of the 12th year is approximately P4,278.24.
In summary:
Redemption value: P50,000
Annual deposits: Approximately P2,670.21
Principal in the fund at the end of the 12th year: Approximately P4,278.24
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Use Heron's Area Formula to find the area of the triangle. (Round your answer to two decimal places.) \[ A=81^{\circ}, b=73, c=39 \]
Let's calculate the area of a triangle using Heron's formula. Heron's formula can be used to calculate the area of a triangle if you know the length of all three sides of the triangle.
When we know the lengths of the sides of a triangle, we can use Heron's formula to calculate the area of the triangle.
Heron's formula is:[tex]$$A=\sqrt{s(s-a)(s-b)(s-c)}$$[/tex]
The semi-perimeter is the half of the triangle's perimeter. It is expressed as the sum of the lengths of all three sides of the triangle divided by two. Let's begin solving the question.
Given that[tex]$a=81^{\circ}$, $b=73$ and $c=39$[/tex]
We need to calculate the area of the triangle,Using the Heron's formula, we have:[tex]\[\begin{aligned} A&=\sqrt{s(s-a)(s-b)(s-c)} \\ s&=\frac{a+b+c}{2} \\ &=\frac{(81^{\circ})+(73)+(39)}{2} \\ &=\frac{193}{2} \end{aligned}\][/tex]
Now, we can use the semi-perimeter of the triangle to calculate the area of the triangle as shown:[tex]\[\begin{aligned} A&=\sqrt{s(s-a)(s-b)(s-c)} \\ &=\sqrt{\frac{193}{2}\left(\frac{193}{2}-81\right)\left(\frac{193}{2}-73\right)\left(\frac{193}{2}-39\right)} \\ &=\sqrt{\frac{193}{2}\cdot\frac{193}{2}\cdot\frac{193}{2}\cdot\frac{193}{2}} \\ &=\sqrt{3744022.25} \\ &=\boxed{1936.96} \end{aligned}\][/tex]
Therefore, the area of the triangle is[tex]1936.96[/tex] square units.
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2- Calculate the two-way shear action for a corner column (450 × 450) mm in a flat plate floor of a span (7.5 x 7.5) m. Find the area of vertical shear reinforcement if required. Assume d = 140 mm. Total applied load qu = 15 kPa (including slab weight). fc = 28 MPa, fy = 420 MPa.
To calculate the two-way shear action for a corner column in a flat plate floor, we need to follow these steps:
1. Determine the applied load on the column:
- Given that the total applied load (qu) is 15 kPa and the span of the flat plate floor is (7.5 x 7.5) m, we can calculate the total applied load (Q) on the column by multiplying the load per unit area (qu) by the floor area (A).
- The floor area (A) is given by A = (7.5 m) x (7.5 m) = 56.25 m².
- Therefore, Q = (15 kPa) x (56.25 m²) = 843.75 kN.
2. Determine the critical section for two-way shear:
- The critical section for two-way shear is located at a distance (d) from the face of the column. In this case, d is given as 140 mm.
3. Calculate the punching shear force (Vc):
- The punching shear force (Vc) can be calculated using the formula Vc = k x √(fc') x b x d, where k is a coefficient, fc' is the effective concrete strength, b is the width of the critical section, and d is the distance from the face of the column to the critical section.
- The coefficient k depends on the shape of the column and the amount of shear reinforcement. For a corner column, the value of k is typically 0.85.
- The effective concrete strength (fc') is calculated as fc' = 0.75 x fc, where fc is the concrete strength.
- The width of the critical section (b) can be determined based on the dimensions of the column and the span of the flat plate floor. In this case, since the column is square with dimensions (450 × 450) mm, the width of the critical section is equal to the side length of the column, which is 450 mm.
- Plugging in the values, Vc = (0.85) x √(0.75 x 28 MPa) x (450 mm) x (140 mm).
4. Determine the area of vertical shear reinforcement (Av) if required:
- To determine if shear reinforcement is required, we need to compare the punching shear force (Vc) to the capacity of the concrete alone (Vn).
- The capacity of the concrete alone (Vn) can be calculated using the formula Vn = φ x √(fc') x b x d, where φ is the resistance factor.
- For two-way shear, the resistance factor (φ) is typically 0.75.
- Plugging in the values, Vn = (0.75) x √(0.75 x 28 MPa) x (450 mm) x (140 mm).
- If Vc > Vn, then shear reinforcement is required. In this case, you need to calculate the area of vertical shear reinforcement (Av) based on the design requirements and the specified yield strength (fy). The design requirements may vary depending on the specific code or design guidelines being followed.
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Establish the identity sin 0 + sin 50 sin 0- sin 50 tan (30) tan 20
The identity is established as sin 0 + sin 50 - sin 0 * sin 50 * tan (30) * tan 20 = 0.766.
To establish the identity, we can simplify each term individually and then combine them using trigonometric identities.
Simplify sin 0:
The sine of 0 degrees is 0.
Therefore, sin 0 = 0.
Simplify sin 50:
We can use the value of sine for 50 degrees from a table or calculator.
Let's assume sin 50 ≈ 0.766.
Simplify sin 0 - sin 50:
Substituting the values from the previous steps, we have:
sin 0 - sin 50 = 0 - 0.766 = -0.766.
Simplify tan (30):
The tangent of 30 degrees can be determined using the value of sine and cosine for 30 degrees.
Let's assume tan 30 ≈ sin 30 / cos 30 ≈ 0.577.
Simplify tan 20:
The tangent of 20 degrees can be determined in a similar way.
Let's assume tan 20 ≈ sin 20 / cos 20 ≈ 0.364.
Now, let's substitute the values into the expression:
sin 0 + sin 50 - sin 0 * sin 50 * tan (30) * tan 20
0 + 0.766 - 0.766 * 0 * 0.577 * 0.364
0 + 0.766 - 0 = 0.766
Therefore, the identity is established as sin 0 + sin 50 - sin 0 * sin 50 * tan (30) * tan 20 = 0.766.
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24. Find ƒ(2+h)-f(2) h a. h + 6 b. 2 c. 2+h d. h² + 2h-1 e. 1 if f(x) = x² + 2x - 1
The function is given as f(x) = x² + 2x - 1.
To calculate ƒ(2+h)-f(2),
we need to first determine the value of ƒ(2+h) and ƒ(2).
ƒ(2+h) is obtained by substituting 2+h in place of x in the function.
f(2 + h) = (2 + h)² + 2(2 + h) - 1= 4 + 4h + h² + 2 + 2h - 1= h² + 6h + 5ƒ(2) is obtained by substituting 2 in place of x in the function.
f(2) = 2² + 2(2) - 1= 4 + 4 - 1= 7
Now, we can find the value of ƒ(2+h) - ƒ(2) as follows:
ƒ(2+h) - ƒ(2) = (h² + 6h + 5) - 7= h² + 6h - 2
Thus, the answer is d. h² + 2h - 1.
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please explain the steps as well, thanks
Evaluate the integral. \[ \int_{1}^{9} \sqrt{y} \ln (y) d y \]
After the steps on Evaluate the integral [tex]\[ \int_{1}^{9} \sqrt{y} \ln (y) d y \][/tex] we get :[tex]\[\frac{2}{3} \cdot 9^{3/2} \ln(9) - \frac{2}{9} \cdot 9^{3/2} - \frac{2}{9}\][/tex]
To evaluate the integral [tex]\(\int_{1}^{9} \sqrt{y} \ln(y) \, dy\)[/tex], we can use integration techniques such as integration by parts.
Let's proceed with the steps to evaluate the integral:
Step 1: Identify the parts of the integrand to be used in the integration by parts technique. In this case, we can choose [tex]\(u = \ln(y)\)[/tex] as the first function and [tex]\(dv = \sqrt{y} \, dy\)[/tex] as the second function.
Step 2: Calculate the derivatives and antiderivatives of the chosen functions:
[tex]\[du = \frac{1}{y} \, dy \quad \text{and} \quad v = \frac{2}{3} y^{3/2}\][/tex]
Step 3: Apply the integration by parts formula:
[tex]\[\int u \, dv = uv - \int v \, du\][/tex]
Plugging in the values, we have:
[tex]\[\int \sqrt{y} \ln(y) \, dy = \left(\ln(y) \cdot \frac{2}{3} y^{3/2}\right) - \int \left(\frac{2}{3} y^{3/2} \cdot \frac{1}{y}\right) \, dy\][/tex]
Simplifying further, we get:
[tex]\[\int \sqrt{y} \ln(y) \, dy = \frac{2}{3} y^{3/2} \ln(y) - \frac{2}{3} \int y^{1/2} \, dy\][/tex]
Step 4: Evaluate the remaining integral:
[tex]\[\int y^{1/2} \, dy = \frac{2}{3} y^{3/2}\][/tex]
Plugging this back into the equation, we have:
[tex]\[\int \sqrt{y} \ln(y) \, dy = \frac{2}{3} y^{3/2} \ln(y) - \frac{2}{3} \cdot \frac{2}{3} y^{3/2} + C\][/tex]
Step 5: Evaluate the integral over the given limits [tex]\([1, 9]\):[/tex]
[tex]\[\left[\frac{2}{3} y^{3/2} \ln(y) - \frac{2}{9} y^{3/2}\right]_{1}^{9}\][/tex]
Plugging in the values, we get:
[tex]\[\left(\frac{2}{3} \cdot 9^{3/2} \ln(9) - \frac{2}{9} \cdot 9^{3/2}\right) - \left(\frac{2}{3} \cdot 1^{3/2} \ln(1) - \frac{2}{9} \cdot 1^{3/2}\right)\][/tex]
Simplifying further, we have:
[tex]\[\frac{2}{3} \cdot 9^{3/2} \ln(9) - \frac{2}{9} \cdot 9^{3/2} - \frac{2}{9}\][/tex]
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Andrew is fishing. If either Andrew is fishing or Ian is swimming then Ken is sleeping. If Ken is sleeping then Katrina is eating. Hence Andrew is fishing and Katrina is eating. B. Andrew is fishing. If either Andrew is fishing of Ian is swimming then Ken is sleeping. If Ken is sleeping then Katrina is eating. Hence Andrew is fishing and Ian is swimming. Test for validity.
The argument is not valid because the conclusion does not necessarily follow from the given premises.
Here, we have,
To test for the validity of the argument, we need to analyze the logical structure of the statements and determine if the conclusion necessarily follows from the premises. Let's break down the argument:
Premises:
Either Andrew is fishing or Ian is swimming → Ken is sleeping.
Ken is sleeping → Katrina is eating.
Conclusion:
Andrew is fishing and Katrina is eating.
To test for validity, we can use a truth table to examine all possible combinations of truth values for the premises and conclusion.
Let's assign the following truth values:
A: Andrew is fishing
I: Ian is swimming
K: Ken is sleeping
Kat: Katrina is eating
The premises can be represented as:
(A ∨ I) → K
K → Kat
The conclusion can be represented as:
A ∧ Kat
Constructing a truth table for these statements, we can check if the conclusion is true in all cases where the premises are true.
A I K Kat (A ∨ I) → K K → Kat A ∧ Kat
T T T T T T T
T T T F T F F
T T F T F T F
T T F F F T F
T F T T T T T
T F T F T F F
T F F T F T F
T F F F F T F
F T T T T T F
F T T F T F F
F T F T T T F
F T F F T T F
F F T T T T F
F F T F T F F
F F F T T T F
F F F F T T F
Based on the truth table, we can see that there are cases where the premises are true (T), but the conclusion is false (F).
Specifically, when Andrew is fishing (A) is false and Katrina is eating (Kat) is false, the conclusion is false.
Therefore, the argument is not valid because the conclusion does not necessarily follow from the given premises.
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In a forest 4/7 of all the trees are conifer and the rest are leaf-bearing. Among the leaf-bearing trees 7/15 are oak and 2/3 of these oaks are new. There are 160 old oaks in the forest. How many trees in all are in the forest? (10 marks) The following are two routes of Toronto Transit Commission (TTC) in the Great Toronto Area in Ontario. Kipling Pape Islington a +3 Royal York 3a-2 20 Old Mill Warden Jane In the above diagrams, all the lengths are in miles. It is noticed that the distance from Kipling to Jane=2 times the distance from Pape to Warden. Find an expression, in terms of a, for the distance from Old Mill to Jane. You MUST present your answer in its simplest form. (10 marks) Question 5 Ashdeep and Simranjeet are selling pencils in the student union store in St. Clair College. They sell boxes of pencils and single pencils. Ashdeep sells 7 boxes of pencils and 22 single pencils. Simranjeet sells 5 boxes of pencils and 2 single pencils. I Ashdeep sells twice as many pencils as Simranjeet. You are required to calculate how many pencils are there in a box. (10 marks) Question 6 In a district election in Brampton between two candidates, Joanna Watson and Kate Simpson. Joanna Watson got 55% of the total valid votes and 20% of the total votes were invalid. If the total number of votes was 7,500, the number of valid votes that Kate Simpson got will be? (10 marks) Question 7 In Test 2 of the OAG 160 Business Essential Mathematics, one of your classmates multiplied a number by 3/5 instead of 5/3, What is the percentage error in his calculation ? (10 marks) Question 8 If 75% of a number is added to 75, then the result is the number itself. What is the number? (10 marks) Question 9 405 sweets were distributed equally among children in such a way that the number of sweets received by each child is 20% of the total number of children. How many sweets did each child receive? (10 marks) Question 10 Your instructor spends 35% of his income on food, 25% on her daughter's education and 80% of the remaining on a condo in Ajax. What percent of your instructor's income is left with? (10 marks)
Answer:
Step-by-step explanation:
Q1) Let T be the total number of trees.
(4/7) T = conifer
⇒ leaf-bearing = T - (4/7) T = (3/7) T
Among the leaf-bearing trees 7/15 are oak
⇒ Oak = (7/15) (3/7) T = (1/5) T
2/3 of these oaks are new.
⇒ 1/3 of these oaks are old
⇒ old oak = (1/3)(1/5) T = (1/15)T
There are 160 old oaks in the forest
⇒ (1/15)T = 160
⇒ T = 160*15 = 2400
There are 2400 trees
Q5)
Let there be x pencils in a box
A = 7x + 22
S = 5x + 2
Also, A = 2S
⇒ 7x + 22 = 2(5x + 2)
⇒ 7x + 22 = 10x + 4
⇒ 10x - 7x = 22 - 4
⇒ 3x = 18
⇒ x = 6
There are 6 pencils in a box
Q6)
Total votes = 7500
20% invalid votes ⇒ 80% valid votes
valid votes = 7500*80% = 6000
Joanna Watson got 55% of the total valid votes
⇒ Kate Simpson got 45% of the total valid votes
K = 6000*45% = 2700 votes
Q7)
[tex]\frac{\frac{5}{3} -\frac{3}{5} }{\frac{5}{3} } *100\%\\\\= \frac{25-9}{15} \frac{3}{5} *100\%\\\\=\frac{16}{25} *100\%\\\\64\%[/tex]
Q8)
Let the number be x
x(75%) + 75 = x
⇒ x - x(75%) = 75
⇒ x(25%) = 75
⇒ x = 75*100/25 = 300
Q9)
Let there be x children
Each child receives the same amount of sweet = 405/x
Also, each child receives x*(20%)
⇒ [tex]\frac{405}{x} = \frac{20}{100}x\\ \\\frac{405*100}{20} = x^{2} \\\\x = \sqrt{\frac{4050}{2} } \\\\x = 45[/tex]
Each child gets x*(20%) sweets
[tex]45\frac{20}{100} \\\\= 9[/tex]
Q10)
Let x be the income
35% x on food
25% x on education
Total expenditure 35% x+ 25% x = 60% x
Remaining income = 40% x
80% on a condo
80% * 40% * x = 32% x
Total spent = 60% x + 32% x = 92% x
Remaining = 8%
A Circle With Centre C Has Equation X2+Y2−10y+20=0 (A) By Completing The Square, Express This Equation In The Form X2+
The equation of the given circle is x² + y² - 10y + 20 = 0. Let us complete the square to express this equation in the form x² + (y - k)² = r², where (h, k) is the center of the circle, and r is the radius.
Step-by-step explanation: We can rearrange the equation as follows:x² + y² - 10y = -20To complete the square, we need to add and subtract the square of half of the coefficient of y. In this case, half of the coefficient of y is -5.
We add and subtract (-5)² = 25 to the equation:x² + y² - 10y + 25 - 25 = -20Add and subtract 25: x² + (y - 5)² = 5²Rearranging the terms gives:x² + (y - 5)² = 25This is now in the form x² + (y - k)² = r². Therefore, the center of the circle is (0, 5) and the radius is 5.
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Find the value of x if A, B, and C are collinear points and B is between A and C.
AB=5,BC=3x+7,AC=5x−2
The value of x is either -1 or 1.
Given that A, B and C are collinear points and B is between A and C, AB = 5, BC = 3x + 7, AC = 5x - 2.
We need to find the value of x.
Let D be the point on the line AB such that CD is parallel to BA.
Then by the basic proportionality theorem,AD / DB = AC / CB ⇒ (5 - x) / x = (5x - 2) / (3x + 7)
Multiplying both sides by (3x + 7) x and simplifying,5 - x = 5x² - 2x ⇒ 5x² - x - 5 = 0
Solving the quadratic equation for x using the quadratic formula we get,x = (- b ± √(b² - 4ac)) / (2a) where a = 5, b = -1 and c = -5
On simplifying this we get, x = -1 or x = 1
Therefore, the value of x is either -1 or 1.
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Evaluate |-3| - |5|.
- 8
- 2
2
8
The answer is:
-2Work/explanation:
Here are two absolute value rules:
[tex]\sf{\mid a\mid =a}\\\\\sf{\mid-a\mid=a}[/tex]
Now evaluate
[tex]\sf{\mid-3\mid-\mid5\mid}\\\sf{3-5}\\\sf{-2}[/tex]
Hence, the answer is -2.