[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
The identity to be used here is [ (a - b)² = a² - 2ab + b² ]
First, try to get the given expression in form of terms on the RHS of the above identity
[tex]\qquad \sf \dashrightarrow \: 4y {}^{2} - 6yz + 9 {z}^{2} [/tex]
[tex]\qquad \sf \dashrightarrow \: (2 {}^{2})( {y}^{2}) - 2(3)(y)(z) + (3 {}^{2} ) {z}^{2} [/tex]
[tex]\qquad \sf \dashrightarrow \: (2y) {}^{2} - (2y)(3z) + (3z) {}^{2} [/tex]
[ it's in form of a² - ab + b² now, add and deduct ab here ]
[tex]\qquad \sf \dashrightarrow \: (2y) {}^{2} - (2y)(3z) + (9z) {}^{2} - (2y)(3z) + (2y)(3z)[/tex]
[tex]\qquad \sf \dashrightarrow \: (2y) {}^{2} - 2(2y)(3z) + (3z) {}^{2} + (2y)(3z)[/tex]
[ Apply the identity ]
[tex]\qquad \sf \dashrightarrow \: (2y - 3z) {}^{2} + (2y)(3z)[/tex]
[tex]\qquad \sf \dashrightarrow \: (2y - 3z) {}^{2} + 6yz[/tex]
[ That's probably the answer, it can't be simplified more ]
To factorize the expression [tex]\sf\:4y^2 - 6yz + 9z^2 \\[/tex], we can use the quadratic formula.
We have the quadratic expression [tex]\sf\:ax^2 + bx + c \\[/tex], where $$\sf\:a = 4$$, $$\sf\:b = -6y $$, and $$\sf\:c = 9z^2 $$.
The quadratic formula states that for any quadratic equation of the form [tex]\sf\:ax^2 + bx + c = 0 \\[/tex], the solutions for $$\sf\:x $$ can be found using the formula:
[tex]\sf\:x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\[/tex]
Applying this formula to our quadratic expression, we get:
[tex]\begin{align}\sf\:x &= \frac{-(-6y) \pm \sqrt{(-6y)^2 - 4(4)(9z^2)}}{2(4)} \\ &= \frac{6y \pm \sqrt{36y^2 - 144z^2}}{8} \\ &= \frac{6y \pm \sqrt{36(y^2 - 4z^2)}}{8} \\ &= \frac{6y \pm 6\sqrt{y^2 - 4z^2}}{8} \\ &= \frac{3}{4}y \pm \frac{3}{4}\sqrt{y^2 - 4z^2}\end{align} \\[/tex]
Thus, the factored form of the expression [tex]\sf\:4y^2 - 6yz + 9z^2 \\[/tex] is [tex]\sf\:(\frac{3}{4}y + \frac{3}{4}\sqrt{y^2 - 4z^2})(\frac{3}{4}y - \frac{3}{4}\sqrt{y^2 - 4z^2}) \\[/tex].
[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]
✨[tex]\large{\textcolor{red}{\underline{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]
systems of equations - substitution method
2a + 7b = 13
8b = 2 - a
By using the substitution method, the value of a is equal to 10 and the value of b is equal to -1.
How to solve this system of equations?In order to solve the given system of equations, we would apply the substitution method. From the information provided in the image attached above, we have the following system of equations:
2a + 7b = 13 .......equation 1.
8b = 2 - a .......equation 2.
From equation 2, we have:
a = 2 - 8b .......equation 3.
By using the substitution method to substitute equation 3 into equation 1, we have the following:
2(2 - 8b) + 7b = 13
4 - 16b + 7b = 13
4 - 9b = 13
9b = 4 - 13
9b = -9
b = -1
a = 2 - 8b
a = 2 - 8(-1)
a = 2 + 8
a = 10.
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Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. Round the answer to at least two decimal places.
[tex]y=e^{x}, y=0,x=0,x=ln(3)[/tex]
Answer:
Step-by-step explanation:
Find the value of 15 - 8-2.
3.5
04
11
Answer:
Step-by-step explanation: Yes, that is correct. The solution to the system of equations is (x, y) = (-7, 0) which is not in the given options, so none of them are the correct answer.
suppose that you have an alphabet of 26 letters. (a) how many possible simple substitution ciphers are there? (b) a letter in the alphabet is said to be fixed if the encryption of the letter is the letter itself. how many simple substitution ciphers are there that leave: (i) no letters fixed? (ii) at least one letter fixed? (iii) exactly one letter fixed? (iv) at least two letters fixed?
A. There are 25^26 possible simple substitution ciphers.
How did we arrive at this assertion?(a) A simple substitution cipher is a substitution of one letter for another. In a simple substitution cipher, each letter in the alphabet can be mapped to one of the 25 other letters. Therefore, for each letter in the alphabet, there are 25 possible substitutions. The total number of possible simple substitution ciphers is the number of possible substitutions for each letter, raised to the power of the number of letters in the alphabet:
25^26 = 25 × 25 × 25 × ... × 25 (26 times)
So, there are 25^26 possible simple substitution ciphers.
(b) (i) If no letters are fixed, then each letter can be mapped to one of the 25 other letters. Therefore, the number of simple substitution ciphers that leave no letters fixed is:
25 × 24 × 23 × ... × 2 × 1 = 25!
(ii) If at least one letter is fixed, then for each letter in the alphabet, there are 24 possible substitutions. Therefore, the number of simple substitution ciphers that leave at least one letter fixed is:
25 × 24 × 23 × ... × 2 × 1 = 25!
(iii) If exactly one letter is fixed, then there are 26 possible letters that could be fixed. For each of these 26 possibilities, there are 24 possible substitutions for each of the remaining 25 letters. Therefore, the number of simple substitution ciphers that leave exactly one letter fixed is:
26 × 24^25 = 26 × 24 × 24 × ... × 24 (25 times) = 26 × 24!
(iv) If at least two letters are fixed, then the number of ways to choose the two fixed letters, multiplied by the number of ways to arrange the other 24 letters, gives the number of ciphers that have exactly two fixed letters. The number of ways to arrange 24 letters is 24!, and there are 26 choose 2 ways to choose two letters from 26:
26 choose 2 × 24! = 325 × 24!
However, this counts ciphers with exactly two fixed letters twice: once for each of the two fixed letters. So, to find the number of ciphers with at least two fixed letters, we need to subtract the number of ciphers with exactly two fixed letters and divide by 2:
(25! - 325 × 24!) / 2 = (25! - 325 × 24!) / 2.
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The top 5% of applicants on a test will receive a scholarship. If the test scores are normally distributed with a mean of 76 and a standard distribution of 12, what is the lowest test score that still qualifies for a scholarship? Use Excel, and round your answer to the nearest integer.
Answer:
96
Step-by-step explanation:
Z-Scores:z-scores are defined as: [tex]z=\frac{x-\mu }{\sigma}[/tex], and this may look a bit confusing at first but by looking at the numerator and then denominator we can get a more understandable definition.
The numerator is first finding the difference between the statistic and the mean. It then divides by the standard deviation, so essentially it's telling us how far the statistic is from the mean in terms of standard deviation.
We can actually rewrite the equation to express this:
[tex]z=\frac{x-\mu }{\sigma}\\\\z\sigma = x-\mu\\\mu + z\sigma=x[/tex]
So in essence, how many standard deviations the statistic is away from the mean.
Now this may seem very off topic compared to what the problem is asking, but we want to convert the top 5% to a z-score. Now let's first convert this top 5% to a percentile. To be in the top 5% you just need to be in the 95th percentile and using technology we can convert this into a z-score which is approximately 1.645.
So this means the 95th percentile is 1.645 standard deviations away and in this case above (since it's positive) from the mean.
The good thing is we know the standard deviation and mean, now let's just apply it: [tex]76+1.645(12)=95.74[/tex]. We now want to round this to the nearest integer of 96, and now we have our answer!
Given f(x) = -3(x + 2), what is the value of f(−7)?
Answer:
f(-7) = 15
Step-by-step explanation:
Subsitute x = - 7 into f (X) = - 3 (x+2)
f(-7) = -3 x (-7+2)
calculate the sum or difference
f(-7) =-3 x (-5)
determine the sign for multiplication or division
f(-7)=3x5
calculate the product or quotient
f(-7)=15
final awnser f(-7)=15
Use the distributive property to evaluate 4(2x - 1) when x = 6.
O44
04
40
O 47
The value of the expression 4(2x - 1) when x = 6 is 44. Option A
What are algebraic expressions?Algebraic expressions are defined as expressions that are made up of coefficients, terms, variables, factors and constants.
They are also identified with arithmetic operations, such as;
AdditionMultiplicationDivisionSubtractionBracketParenthesesFrom the information given, we have that;
4(2x - 1)
expand the bracket, we get;
8x - 4
Substitute the value of x as , we have;
8(6) - 4
Multiply
48 -4
Subtract the values
44
Hence, the value is 44
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Which statement is true about f(x) +2=1/6|x-3|?
Answer: The range of f(x) is [tex]f(x)\geq -2[/tex]
Step-by-step explanation:
The given function is
[tex]f(x)+2=\frac{1}{6}|x-3|[/tex]
It can be written as
[tex]f(x)=\frac{1}{6}|x-3|-2[/tex] ...(1)
The function is in the form of
[tex]g(x)=a|x-h|+k[/tex] ...(2)
Where, a is scale factor and (h,k) is vertex of the graph.
On comparing (1) and (2), we get
[tex]a=\frac{1}{6}[/tex]
[tex]h=3[/tex]
[tex]k=-2[/tex]
The value of a is [tex]\frac{1}{6}[/tex] . So, the graph compressed vertically. The value of a is positive, therefore the graph of f(x) opens upward.
We know the absolute value is always greater than or equal to 0.
[tex]|x-3|\geq 0[/tex]
[tex]\frac{1}{6}|x-3|\geq \frac{1}{6}(0)[/tex]
[tex]\frac{1}{6}|x-3|\geq 0[/tex]
[tex]\frac{1}{6}|x-3|-2\geq 0-2[/tex]
[tex]f(x)\geq -2[/tex]
hope you've understood...
Are girls better at spelling than boys? An SRS of 200 boys and an SRS of 150 girls was administered a
spelling test. The average score for boys was 48.4 with standard deviation 12.96. The girls had an average score
of 48.9 with standard deviation 11.85. Construct and interpret a 95% confidence interval to estimate the true
mean difference in the scores of the spelling test between boys and girls. Does this interval suggest a
difference? Explain.
Based on the information, it should be noted that the difference between the scores of the girls and boys aren't significant.
How to explain the informationFrom the information, the average score for boys was 48.4 with standard deviation 12.96 and the girls had an average score
of 48.9 with standard deviation 11.85.
The computed upper limit = 2.1119
The computed lower limit = -3.1119
Since the interval (-3.1119, 2.1119) contains the between the value 0 thus the difference of boys and girls scores isn't significant.
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find the value of each variable provide proofs
Answer:
both sides are equal to: [tex]8\sqrt{2}[/tex]
Step-by-step explanation:
Trigonometric Functions:Trigonometric functions are defined as the ratios between sides based on some angle. There's three main trig functions you want to remember which are defined as:
[tex]sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\\\\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\\\\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}[/tex]
and then there are three other trig functions, which are simply the reciprocals:
[tex]csc(\theta)= \frac{\text{hypotenuse}}{\text{opposite}}=\frac{1}{sin(\theta)}\\\\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}=\frac{1}{cos(\theta)}\\\\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}=\frac{1}{tan(\theta)}[/tex]
Now in our case we have the angle of 45 degrees and we have the hypotenuse with length 16, and we want to either solve for the opposite (x) or adjacent side (y)
Let's solve for the adjacent side, in which case we want a trig function defined using the hypotenuse and adjacent side, which is our cosine function.
[tex]cos(45)=\frac{y}{16}\implies cos(45)*16=y[/tex] and from here we can use a calculator or the unit circle which gives us an exact value. Using the unit circle, we can determine that: [tex]cos(45)=\frac{\sqrt{2}}{2}[/tex] and plugging that into our equation we get: [tex]y=\frac{\sqrt{2}}{2}*16=8\sqrt{2}[/tex] and we can use a calculator to approximate this: 11.313
Now we can also use the trig functions to find "x", which is the opposite side. In this case we want to use a trig function which is defined using the opposite and hypotenuse side, which is our sine function: [tex]sin(45)=\frac{x}{16}\implies sin(45)*16=x[/tex] and the cool thing about this, is that sin(45) and cos(45) are actually equal, so our "x" and "y' side are exactly equal: [tex]x=8\sqrt{2}[/tex]
We can verify this using the Pythagorean Theorem: [tex](8\sqrt{2})^2+(8\sqrt{2})^2=16^2\\\\(8^2*\sqrt{2}^2)+(8^2+\sqrt{2}^2)=256\\\\(64*2)+(64*2)=256\\\\128+128=256\\\\256=256[/tex]
A group of students stood in a circle to play a game. The circle had a diameter of 22 meters. Which measurement is closest to the area of the circle in meters?
The measurement closest to the area of the circle in meters is 379.94 sq. meters.
What is area?An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.
The word "area" refers to a free space. A shape's length and width are used to compute its area. The majority of forms and things have corners and edges. When computing the area of a certain form, both the length and width of these edges are taken into account.
Given that,
diameter = 22m
radius = 22/2 = 11m
The area of the circle is:
A = πr²
Substituting the value of π = 3.14 and r = 11 we have:
A = (3.14)(11)²
A = 379.94
Hence, the measurement closest to the area of the circle in meters is 379.94 sq. meters.
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Selma wants to take her money out of the bank account with simple interest rate of 5% annually and put it in another account with compound interest rate of 8% compounded quarterly if the bank says closing the current account has a penalty of 5% at least how many years she let the money stay at the new account in order to recover the penalty paid using extra interest of the second?
Selma needs to leave the money in the new account for at least one year, option A. 1 to recover the penalty paid.
How did we get this assertion?Assuming Selma wants to recover the penalty paid using the extra interest earned in the new account, she needs to calculate the amount of time it will take for the new account's interest to make up for the penalty paid.
Let P be the initial amount of money in Selma's current account. The penalty for closing the account is 5% of P, which is 0.05P.
If Selma leaves the money in the current account for one year, she will earn simple interest of 5% of P, which is 0.05P. So at the end of the year, the balance in the current account will be P + 0.05P = 1.05P.
If Selma transfers the money to the new account, it will earn compound interest of 8% per year, compounded quarterly. The quarterly interest rate is 8%/4 = 2%.
After one quarter, the balance in the new account will be:
P*(1 + 2%/4) = P*1.02
After two quarters, the balance will be:
P1.02(1 + 2%/4) = P*1.02^2
After three quarters, the balance will be:
P1.02^2(1 + 2%/4) = P*1.02^3
After four quarters (i.e., one year), the balance will be:
P1.02^3(1 + 2%/4) = P*1.02^4
So at the end of one year, Selma will have 1.05P in the current account and P*1.02^4 in the new account.
To recover the penalty of 0.05P, Selma needs the balance in the new account to be at least 1.05P. That is,
P*1.02^4 >= 1.05P
Simplifying, we get:
1.02^4 >= 1.05
Taking the fourth root of both sides, we get:
1.02 >= 1.05^(1/4)
Using a calculator, we get:
1.02 >= 1.01298
So Selma needs to leave the money in the new account for at least one year to recover the penalty paid.
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Use the scale to help you solve the equation and find the value of x?
A 80 kg monkey climbs a 15 meter tree in half a minute. What is the magnitude of power the monkey demonstrated?.
The demonstration of magnitude of power given by 80 kg monkey by climbing a 15 meter tree in half a minute is equal to option b. 392 J/S.
Weight 'm' of the monkey is equal to 80kilogram
Height 'h' of the tree is equal to 15 meter
Time 't' taken by monkey to climb a tree = half a minute
= 30 minutes
g = 9.8 m/s²
Magnitude of power = ( m × g × h )/ t
⇒ Magnitude of power = ( 80 × 9.8 × 15 )/ 30
⇒ Magnitude of power = 784 / 2
⇒ Magnitude of power = 392 J/S
Therefore, the magnitude of the power for the given weight , displacement and time is equal to option b. 392 J/S.
The above question is incomplete, the complete question is:
A 80 Kg monkey climbs a 15 meter tree in half a minute. What is the magnitude of power the monkey demonstrated?
a. 13.1 J/S
b. 392 J/S
c. 784 J/s
d. 11760 J/s
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PLS HELP ASAP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:15n + 3 = 153
Step-by-step explanation:
15n + 3 = 153
product of 15 and a number n is 15n, and 3 more gets the plus 3, and it said all of this is 153
n=10 also as subtract 3 both sides then get 15n=150 get n by itself by dividing both sides by 15 and get n=10
Use the coordinates below to determine if AABC and ADEF are congruent.
AABC: A(2, -8), B(-5, -2), C(-7, 3); ADEF: D(-9, 7), E(-11, 12), F(-2, 1)
AB=
DE=
BC=
EF=
AC=
DF=
Are the angles congruent? If yes, explain your reasoning and write a congruency statement.
No The triangles' are not congruent
How to determine if the triangles are congruentThe length of the corresponding sides should be equal and the length of sides is gotten using the equation:
The length of line in an ordered pair is calculated using the formula
d = √{(x₂ - x₁)² + (y₂ - y₁)²}
where
d = distance between the points
x₂ and x₁ = points in x coordinates
y₂ and y₁ = points in y coordinates
distance between points A(2, -8), B(-5, -2), C(-7, 3); and ΔDEF: D(-9, 7), E(-11, 12), F(-2, 1) are calculated as follows:
AB =√{(-2 + 5)² + (-8 + 2)²} = 6.71
DE = √{(-9 + 1)² + (7 - 12)²} = 9.43
BC = √{(-5 + 7)² + (-2 - 3)²} = 5.39
EF = √{(-11 + 2)² + (12 - 1)²} = 14.21
AC = √{(2 + 7)² + (-8 - 3)²} = 14.21
DF = √{(-9 + 2)² + (7 - 1)²} = 9.22
Examining the figures showing the side lengths shows that the two triangles are not congruent
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When rolling 2 number cubes what is the probability of rolling a 6
Answer:
Step-by-step explanation:
To find the probability of rolling a 6 on two number cubes, we need to calculate the number of favorable outcomes (rolling a 6 on either or both of the cubes) divided by the total number of possible outcomes.
Each cube has 6 sides, numbered 1 through 6. So, the total number of possible outcomes when rolling two cubes is 6 x 6 = 36.
The number of favorable outcomes is 2, because there are two ways to roll a 6: either rolling a 6 on the first cube and a 6 on the second cube, or rolling a 6 on one of the cubes.
So, the probability of rolling a 6 is:
favorable outcomes / total outcomes = 2 / 36 = 1/18
Therefore, the probability of rolling a 6 on two number cubes is 1/18 or approximately 5.56%.
The regular price of an Apple Watch is $199. The watch is on sale for 15% off. Which expression is equal to the sale price in dollars, of the Apple Watch
Answer: $169.15
Step-by-step explanation: 15 percent of 199 is 29.85, and 199-29.85 is 169.15
a ferris wheel at a carnival has a diameter of 62 feet. suppose a passenger is traveling 6 miles per hour. Find the angular speed in radians per minute. Find the number of revolutions the wheel makes per hour.
On solving the provided question, we can say that relation between tangential speed and angular speed, v = r * w; v: tangential speed; r: radius; w: angular speed
what is radius?The length of a circle or sphere, in more contemporary use, is the same as its radius in classical geometry, which is one of the line segments from its center to its circumference. The Latin word radius, which also refers to the spokes of a wagon wheel, gave rise to the term. The distance a circle's center is from any point on its perimeter is its radius. Usually, "R" or "r" is used to indicate it. A radius is a line segment that has one endpoint in the center and one on the circumference of a circle. Circular diameter equals radius The diameter of a circle is the segment that traverses its center and has ends that are on the circle.
relation between tangential speed and angular speed
v = r * w
v: tangential speed
r: radius
w: angular speed
the radius is
r = d/2
d is the diameter
[tex]v = (d * w)/2\\w = 2*v/d\\w = (2*9 mile/h)/(58 feet)\\w = (2*9 mile/h)/(0.011 miles) = 1636 rad/h\\w = 1636/60 = 27.2 rad/min\\[/tex]
the angular speed is in radians,
[tex]n = v/(π*d)[/tex]
n = (9 mile/h)/(π*0.011 mile) = 260 rev/h
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Sharon wants to buy a shirt that costs $50. The sales tax is 5%. How much is the sales tax? What is her total cost for the shirt
Answer: The sales tax is $2.5 and her total cost of the shirt is $47.5.
Step-by-step explanation:
help please asapppppp
Answer: The answers are both B and C
A recycling bin is in the shape of a rectangular box. Find the height of the box if its length is 16 ft, its width is 7 ft, and its surface area is 500 ft2. (In the figure, h=height. Assume that the given surface area includes that of the top lid of the box.)
Answer:
Step-by-step explanation:
The surface area of a rectangular box can be calculated as follows:
Surface area = 2lw + 2lh + 2wh
Where l is the length, w is the width, and h is the height.
We are given the surface area (500 ft2) and the length and width (16 ft and 7 ft respectively), so we can use these values to solve for the height:
500 = 2 * 16 * 7 + 2 * 16 * h + 2 * 7 * h
500 = 224 + 32h + 14h
500 = 238h + 224
276 = 238h
h = 276 / 238
h = 1.16 ft
So the height of the box is approximately 1.16 feet.
Use the formula for continuous compounding to compute the balance in the account after 1, 5, and 20 years. Also, find the APY for the account. A $1000 deposit in an account with an APR of 3%. The balance in the account after 1 year is approximately
The APY for the account would be 3.045%.
What is continuous compounding?
The formula for continuous compounding is given by:
A = P[tex]e^{(rt)}[/tex]
where:
A = final amount
P = principal amount
e = the mathematical constant e (approximately 2.71828)
r = annual interest rate
t = time in years
In this case, P = $1000, r = 0.03 (since APR is given), and t = 1 year.
Using the formula, we have:
A = 1000[tex]e^{(0.03*1)}[/tex]
A ≈ $1030.45
So the balance in the account after 1 year is approximately $1030.45.
To find the balance after 5 and 20 years, we simply need to plug in the corresponding values for t:
For t = 5:
A = 1000[tex]e^{(0.03*5)}[/tex]
A ≈ $1160.92
For t = 20:
A = 1000[tex]e^{(0.03*20)}[/tex]
A ≈ $1806.11
To find the APY, we use the formula:
APY = (1 + r/n)ⁿ- 1
where:
n = number of compounding periods per year
Since this is continuous compounding, we take the limit as n approaches infinity:
APY = eʳ - 1
APY = [tex]e^{0.03}[/tex] - 1
APY ≈ 0.03045 or 3.045%
Hence, the APY for the account would be 3.045%.
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You are given the yearly interest earned from a total of $18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.
Yearly Interest: $579
Rate 1: 2.75%
Rate 2: 4.75%
(dollars invested at 2.75%, dollars invested at 4.75%) =
The amount of investment at a rate of 2.75% and 4.75% will be $13,800 and $4,200m, respectively.
What is simple interest?Simple interest is the concept that is used in many companies such as banking, finance, automobile, and so on.
The formula for interest is written as,
I = (PRT)/100
Where P is the principal, R is the rate of interest, and T is the time.
Let 'x' be the amount of investment at a rate of 4.75%. Then the amount of investment at a rate of 2.75% is ($18,000 - x). Then the equation is given as,
(18,000 - x) × 0.0275 + 0.0475x = 579
495 - 0.0275x + 0.0475x = 579
0.02x = 84
x = $4,200
Then the value of ($18,000 - x) is given as,
$18,000 - x = $18,000 - $4,200
$18,000 - x = $13,800
The amount of investment at a rate of 2.75% and 4.75% will be $13,800 and $4,200m, respectively.
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A salesperson's weekly paycheck is 25% more than a second salesperson's paycheck. The two paychecks total $1225. Find the amount of each paycheck. (Round your answers to the nearest cent.) first salesperson's paycheck $ second salesperson's paycheck
On solving the provided question we can say that the equation is p + 1.25p = 1225 => 2.25p = 1225 => p = 544.44
What is equation?A formula is a formula that connects two statements and uses the equal sign (=) to indicate equality. A formula that proves the equality of two formulas is called an equation in algebra. For example, in the expression 3x + 5 = 14, the equal sign separates the variables 3x + 5 and 14. The relationship between the two sentences on either side of the letter is represented by a mathematical formula. In many cases, only one variable can also act as a symbol. For example, 2x – 4 = 2.
the equation is
p + 1.25p = 1225
2.25p = 1225
p = 544.44
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Eighteen low-fat wheat crackers have 3 grams of fat. How many grams of fat do 63 of these crackers have?
Answer:
10.5
Step-by-step explanation:
18/3 = 63/x
18 * x = 63
18 * 3.5 = 63
3 * 3.5 = 10.5
Evaluate the following expressions. a) 10^15/10^12.
b) 10^1/10^0
c) 10^5+10^-3
d) 10^5-10^3
(^ means to the power of)
On solving the provided question we can say that the expressions are
a) 1000
b 10
c) 100000.001
d) 99000
What is expression?In mathematics, it is pοssible to multiply, divide, add, or remove. The construction of an expression is as follows: Expressiοn, number, and mathematical operatοr Numbers, variables, and functions are the building blocks of a mathematical expressiοn (such as addition, subtraction, multiplicatiοn or division etc.)
Expressiοns and phrases can be contrasted. Any mathematical statement with variables, numbers, and an arithmetic οperation between them is called an expression or an algebraic expressiοn. For instance, the expressiοn 4m + 5 has the terms 4m and 5 as well as the variable m of the supplied expressiοn, all of which are separated by the arithmetic sign +.
The expressiοns are
a)10¹⁵/10¹² = 10³ = 1000
b)10¹/10⁰ = 10/1 = 10
c)10⁵ + 10⁻³ = [tex]\dfrac{10^8 +1}{100}[/tex]= 100000.001
d)10⁵ - 10³ = 100000 - 1000 = 99000
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40°
Find the value of x.
X
190°
x = [?]°
The value of x for the given circle will be going to be 50°.
What is a circle?A circle is a geometrical figure which becomes by plotting a point around a fixed point by keeping a constant distance.
In our daily life, we always see circle objects for example our bike wheel.
Area of circle = πr² and the perimeter of circle = 2πr
where r is the radius of the circle.
Angles of Intersecting Chords Theorem
When two chords cross inside of a circle, the resulting angle's measure is equal to the product of the lengths of the arcs it intercepts and its vertical angle, divided by two.
So, angle x = (190° + 40°)/2 = 115°
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Differentiate y=sin x using first principles
Answer:
[tex]\dfrac{\text{d}y}{\text{d}x}=\cos(x)[/tex]
Step-by-step explanation:
Differentiating from First Principles is a technique to find an algebraic expression for the gradient at a particular point on the curve.
[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Differentiating from First Principles}\\\\\\$\text{f}\:'(x)=\displaystyle \lim_{h \to 0} \left[\dfrac{\text{f}(x+h)-\text{f}(x)}{(x+h)-x}\right]$\\\\\end{minipage}}[/tex]
The point (x + h, f(x + h)) is a small distance along the curve from (x, f(x)).
As h gets smaller, the distance between the two points gets smaller.
The closer the points, the closer the line joining them will be to the tangent line.
To differentiate y = sin(x) using first principles, substitute f(x + h) = sin(x + h) and f(x) = sin(x) into the formula:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x+h)-\sin(x)}{(x+h)-x}\right][/tex]
Use the sin addition formula to expand sin(x + h).
[tex]\boxed{\sin(A + B) = \sin A \cos B + \cos A \sin B}[/tex]
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{(x+h)-x}\right][/tex]
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x)\cos(h)-\sin(x)+\cos(x)\sin(h)}{h}\right][/tex]
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x)(\cos(h)-1)+\cos(x)\sin(h)}{h}\right][/tex]
Separate the sin(x) and cos(x) terms into two fractions:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x)(\cos(h)-1)}{h}+\dfrac{\cos(x)\sin(h)}{h}\right][/tex]
When h gets really small, we can use the small angle approximation to rewrite cos(h).
[tex]\boxed{\cos (h) \approx 1-\dfrac{1}{2}h^2}[/tex]
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x)\left(1-\dfrac{1}{2}h^2-1\right)}{h}+\dfrac{\cos(x)\cdot h}{h}\right][/tex]
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{\sin(x)\left(-\dfrac{1}{2}h^2\right)}{h}+\dfrac{\cos(x)\cdot h}{h}\right][/tex]
Cancel the common factor, h:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[-\dfrac{1}{2}h\sin(x)+\cos(x)\right][/tex]
As h → 0, the first term → 0:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\cos(x)[/tex]
Hello can you help me solve
Answer:
(p o q) = 54
(q o p) = 68
Step-by-step explanation:
Another way to write is (p o q)(3) is p(q(3)). As you can see, you start with the inner function and 3 is an input in the q(x) function. Then, the output of this becomes the input of p(x) (the outer function), which yields a final output and the answer:
[tex]p(x) = 4x + 2\\q(x)=5x-2\\\\p(q(3))\\\\q(3)=5(3)-2=13\\p(13)=4(13)+2=54[/tex]
You do the same process for (q o p)(3), which can be written as q(p(3)), where you start with p(3) and then plug in this result as the input for q(x):
[tex]p(x)=4x+2\\q(x)=5x-2\\\\q(p(3))\\\\p(3)=4(3)+2=14\\q(14)=5(14)-2=68[/tex]
