Please help you guys, the AP calc an exam coming soon and I need to know how to do this, much appreciated.
Answers
To be continuous at x=0, you need three things:
1. f(0) must be defined
2. limit as x –> 0 must exist
3. f(0) = limit as x –> 0
f(0) in this case is the following:
[tex]f(0) = k + 2 \ln(0+e^{0+1}) = k + 2 \ln(e) = k + 2[/tex]
We have to pause on this, since this by itself cannot give us the value of k.
For the limit, you need to make sure the limit from the left matches the limit from the right.
[tex]\lim\limits_{x\to 0^+} k+2\ln(x+e^{x+1}) = k+2[/tex]
[tex]\lim\limits_{x\to 0^-} \frac{\sin(7x)}{2x} =\frac{7}{2}[/tex]
The only way this limit can exist is if [tex]k+2 = \dfrac{7}{2}[/tex], which means [tex]k = \dfrac{3}{2}[/tex].
Using that k-value also allows f(0) = 7/2, which makes your function continuous.
If you need to know how to make that second limit work out to 7/2, there are two steps to take.
The first is to bring the 1/2 out front, since it's a constant factor inside the limit.
[tex]\lim\limits_{x\to 0^-} \frac{\sin(7x)}{2x} =\frac{1}{2}\cdot\lim\limits_{x\to 0^-} \frac{\sin(7x)}{x}[/tex]
The second is to manipulate the what you're taking the limit of to turn it into a "sin(u)/u" situation. To do this, we'll multiply by 7/7 inside the limit:
[tex]\frac{1}{2}\cdot\lim\limits_{x\to 0^-} \frac{\sin(7x)}{x}=\frac{1}{2}\cdot\lim\limits_{x\to 0^-} \frac{{\boldsymbol7\cdot{}}\sin(7x)}{{\boldsymbol7\cdot{}}x}[/tex]
We'll then factor out the 7 in the numerator like we did with the 1/2:
[tex]=\frac{7}{2}\cdot\lim\limits_{x\to 0^-} \frac{\sin(7x)}{7x}[/tex]
And now with a quick u-substitution, we'll let u = 7x and x->0 is the same as u->0, we have
[tex]=\frac{7}{2}\cdot\lim\limits_{u\to 0^-} \frac{\sin(u)}{u}[/tex]
This is useful because a good limit to know in calculus is that [tex]\lim\limits_{u\to 0} \frac{\sin(u)}{u}=1[/tex].
Related Questions
What is the exact volume of the figure in terms of Pi?
Answers
The volume of the cone in terms of π is 168π in³
What is volume of a cone?A cone is a three-dimensional shape in geometry that narrows smoothly from a flat base (usually circular base) to a point(which forms an axis to the centre of base) called the apex or vertex.
The volume of a cone is one- third of a volume of a cylinder. Therefore;
The volume of a cone is expressed as;
V = 1/3 πr²h
where r is the radius and h is the height
r = 6 in
h = 14 in
V = 1/3 × π × 6² × 14
V = 504π/3
V = 168π in³
Therefore the volume of the cone in terms of π is 168π in³
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14. Sami lives directly east of the park. The football field is directly south of the park. The library sits on the line formed between Sami’s triangle formed by her home, the park, and the football field could be drawn. The library is 4 miles from her home. The football field is 18 miles from the library.
How far is the library from the park?
How far is the park from the football field?
Answers
The library is 6.3 miles away from the park. the library and football fields are 11.8 miles away
Derive the equation for the ellipse centered at the origin with foci (0, ±2c) and major vertices (0, ±2a).
Answers
The equation of the ellipse is x²/a² + y²/(9a²) = 1
Given data ,
The ellipse centered at the origin with foci (0, ±2c) and major vertices are given as (0, ±2a)
From the given information, we have:
Center (h, k) = (0, 0)
Foci: (0, ±2c)
Major Vertices: (0, ±2a)
The distance formula between the center and each focus is given by:
c² = a² - b²
We can rewrite the equation using the given information as follows:
c = 2c (distance from center to focus)
Therefore, we have:
4c² = a² - b²
Now, let's consider the distances from the center to each vertex:
a = 2a (distance from center to vertex)
Therefore, we have:
4a² = a² - b²
Rearranging the equation, we get:
3a² = b²
Now, substituting this value of b² in terms of a² into the standard form equation for an ellipse, we have:
x²/a² + y²/(3a²) = 1
Simplifying further, we get:
x²/a² + y²/(9a²) = 1
Hence , the equation for the ellipse centered at the origin is
x²/a² + y²/(9a²) = 1
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look at the attached picture to see want this post is asking for.
U dont have to sketch
1. f(x) = x^2 + 6x +2
2. f(x) = 2 - 5x= 2x^2
3. f(x) = -0.25x^2 + 8x - 2
4. f(x) = 4x^2 + 3x -5
Answers
The directions of the parabola are:
1. f(x) = x² + 6x +2 ⇒ Open up2. f(x) = 2 - 5x + 2x² ⇒ Open up3. f(x) = -0.25x² + 8x - 2 ⇒ Open down4. f(x) = 4x² + 3x -5 ⇒ Open upOther solutions are below
Calculating the direction of the parabolaThe equation of a parabola is represented as
y = ax² + bx + c
If a > 0, then the parabola will open up
If a < 0, then the parabola will open down
Using the above as a guide, we have the following:
1. f(x) = x² + 6x +2 ⇒ Open up2. f(x) = 2 - 5x + 2x² ⇒ Open up3. f(x) = -0.25x² + 8x - 2 ⇒ Open down4. f(x) = 4x² + 3x -5 ⇒ Open upCalculating the vertex of the parabolaThis is calculated as (h, k)
Where
h = -b/2a
So, we have
1. f(x) = x² + 6x +2
h = -6/2(1) = -3f(-3) = (-3)² + 6(-3) +2 = -7Vertex = (-3, -7)Axis of symmetry: x = -32. f(x) = 2 - 5x + 2x²
h = -(-5)/(2*2) = 1.25f(1.25) = 2 - 5(1.25) + 2(1.25)² = -1.125Vertex = (1.25, -1.125)Axis of symmetry: x = 1.253. f(x) = -0.25x² + 8x - 2
h = -8/(2*-0.25) = 16f(16) = -0.25(16)² + 8(16) - 2 = 62Vertex = (16, 62)Axis of symmetry: x = 164. f(x) = 4x² + 3x -5
h = -2/(2*4) = -0.25f(-0.25) = 4(-0.25)² + 3(-0.25) -5 = -5.5Vertex = (-0.25, -5.5)Axis of symmetry: x = -0.25Read more about quadratic functions at
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expand (√2x+2y-√3z)^2 using identities fast
Answers
The Expansion of (√2x + 2y - √3z)² is 2x + 4y + 3z - 2√6xy - 2√6xz - 4√3yz.
To expand the expression (√2x + 2y - √3z)², we can use the following identity:
(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc
In this case, we can identify a = √2x, b = 2y, and c = -√3z. So we have:
(√2x + 2y - √3z)²
= (√2x)² + (2y)² + (-√3z)² + 2(√2x)(2y) + 2(√2x)(-√3z) + 2(2y)(-√3z)
Simplifying, we get:
(√2x + 2y - √3z)² = 2x + 4y + 3z - 2√6xy - 2√6xz - 4√3yz
Therefore, (√2x + 2y - √3z)² = 2x + 4y + 3z - 2√6xy - 2√6xz - 4√3yz.
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What Z-score value separates the top 70% of a normal distribution from the bottom 30%? a. z=0.52
c. z=-0.52 b. z=0.84 d. 2= -0.84 10.
Answers
The area to the left of the z-score corresponds to the cumulative probability up to that point. The correct answer is b. z=0.84.
To find the z-score that separates the top 70% from the bottom 30%, we need to look up the corresponding percentile in a standard normal distribution table.
The area to the left of the z-score corresponds to the cumulative probability up to that point. For example, if we look up a z-score of 0.84 in the table, we find that the area to the left of that value is 0.7995, or 79.95%.
Since we want to find the value that separates the top 70% from the bottom 30%, we subtract 0.30 (or 30%) from 1.00 to get 0.70 (or 70%). We then find the z-score that corresponds to that area, which is approximately 0.84.
Therefore, the answer is b. z=0.84.
To find the Z-score value that separates the top 70% of a normal distribution from the bottom 30%, you need to look for the Z-score corresponding to the 70th percentile. In this case, the correct answer is:
b. z=0.52
Here's a step-by-step explanation:
1. Determine the percentile you're looking for, which is the 70th percentile (separating the top 70% from the bottom 30%).
2. Consult a Z-score table or use a calculator to find the Z-score corresponding to the 70th percentile.
3. The Z-score for the 70th percentile is approximately 0.52.
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Use the formula to find the surface area of the figure. Show your work.
Answers
Answer:
502.65yd^2
Step-by-step explanation:
To find the surface area of a cylinder, the surface area is 2(pi)(r)(h)+2(pi)(r^2)
Plug in all values into the formula.
We will use pi as 3.14.
Because the diameter is 10, the radius will be 5.
2(3.14)(5)(11)+2(3.14)(25)
Therefore, the surface area of the cylinder is about 502.65 yd^2.
Scientific Notation Tutorial - Part 2- Level H
The average diameter of an atom's nucleus is
about 1 x 10-14 meter. The diameter of a proton is
about 1 x 10-15 meter.
The diameter of a proton times 10 raised to what power is equivalent to
the diameter of a nucleus?
Proton
10-15 x 10
DONE
10-14
Nucleus
Answers
To determine how many times larger the diameter of a nucleus is than the diameter of a proton, we need to divide the diameter of the nucleus by the diameter of the proton:
(1 x 10^-14 meter) / (1 x 10^-15 meter) = 10
Therefore, the diameter of a nucleus is 10 times larger than the diameter of a proton.
Alternatively, we can say that the diameter of a proton times 10 raised to the power of 1 is equivalent to the diameter of a nucleus:
(1 x 10^-15 meter) x 10^1 = 1 x 10^-14 meter
So, the answer is 1.
I’m so confused Someone help pls
Answers
Step-by-step explanation:
b = 37 degrees due to opposite angles of crossing lines
c = a 37 + c = 180 degrees so c = a = 143 degrees
what is the volume of the solid
Answers
The calculated volume of the solid is 335.10 cubic feet
Calculating the volume of the solidFrom the question, we have the following parameters that can be used in our computation:
Radius = 4 feet
Height = 10 fet
The angle is given as
Angle = 120 degrees
Using the above as a guide, we have the following:
Volume = (360 - Angle)/360 * πr²h
Substitute the known values in the above equation, so, we have the following representation
Volume = (360 - 120)/360 * π * 4² * 10
Evaluate
Volume = 335.10
Hence. the volume of the solid is 335.10 cubic feet
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13th term of the geometric sequence 1, 2, 4
Answers
Answer:
The 13th term of the geometric sequence is 4096---------------------
First, find the common ratio:
r = 2/1 = 4/2 = 2Next, use the formula for the nth term of a geometric sequence:
aₙ = a₁ * rⁿ⁻¹ where a₁ is the first term and r is the common ratioPlugging in the values we have:
a₁₃ = 1 * 2¹³⁻¹ = 1 * 2¹² = 4096URGENT!! Will give brainliest:)
Suppose the line of best fit is drawn for some data points.
If the mean of the x-coordinates of the points is 8, and the mean of the y-coordinates of the points is -10, the line must pass through which of these points?
A. (-10, 8)
B. (-8, 10)
C. (10, -8)
D. (8, -10)
Answers
Answer:
the mean is 8 for x
8 = x + 2x / 2
16 = 3x
x = 16/3
x = 5.3
therefore the coordinate of x are ( 5.3,10.6)
for the y coordinate
-10= y +2y /2
-20 = 3y
y = -6.6
The coordinate of y are ( -6.6, - 13.3 )
A different airplane travels at a constant speed of 500 miles per hour.
• Write an equation that models the number of miles flown, y, by this airplane given x hours.
• Identify the dependent variable and describe why it is the dependent variable.
• How many miles will this airplane travel in 3.5 hours?
Enter your equation, description, and answer in the box provided.
Answers
The Airplane travel 1750 miles in 3.5 hours.
The equation that models the number of miles flown by the airplane given x hours is:
y = 500x
where y is the dependent variable, representing the number of miles flown, and x is the independent variable, representing the number of hours flown.
The dependent variable is y because it depends on the independent variable x. The number of miles flown by the airplane is determined by the number of hours it has flown. The longer the airplane flies, the more miles it will cover. Therefore, the number of miles flown is a function of the number of hours flown, and it is the dependent variable.
To find how many miles the airplane will travel in 3.5 hours, we substitute x = 3.5 into the equation:
y = 500x
y = 500(3.5)
y = 1750
Therefore, the airplane will travel 1750 miles in 3.5 hours.
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There are $4 each and 40 of them. How much money would you save with a deal of buy 4 get 1 free as opposed to buy 9 get 2 free
Answers
Assuming that the regular price for each item is $4 and there are 40 items in total, the deal of buy 4 get 1 free would save $32 compared to the deal of buy 9 get 2 free.
To calculate the amount saved with each deal, we first need to determine how many free items we would receive with each deal. With the buy 4 get 1 free deal, for every 5 items purchased, 1 item is free. This means that we would receive a total of 8 free items (40/5 = 8). With the buy 9 get 2 free deal, for every 11 items purchased, 2 items are free. This means that we would receive a total of 7.27 free items (40/11*2 = 7.27).
With the buy 4 get 1 free deal, we would pay for a total of 32 items (40 - 8 free items). This would cost a total of $128 (32 x $4). With the buy 9 get 2 free deal, we would pay for a total of 32.73 items (40 - 7.27 free items). This would cost a total of $130.92 (32.73 x $4). Therefore, the buy 4 get 1 free deal would save us $2.92 compared to the buy 9 get 2 free deal.
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what is the area of a circle whose circumference is87.92 cm? take pi 3 .14
Answers
Answer:
615.75cm
Step-by-step explanation:
The circumference of a circle is 2r×pi
If the circumference is 87.92cm, we can work out the radius
2×r×pi=87.92cm
r=87.92÷(2×pi)
r=14
Now we can work out the area using the formula r²×pi
A=r²×pi
A=14²×3.14
A=196×3.14
A=615.75
It’s proportions
Need help with first four
Answers
It will take Alice 65 minutes to finish mowing the lawn.
How long will it take Alice to finish mowing?To mow a 6 x 32 m strip of lawn, Alice covered an area of:
= 6 x 32
= 192 square meters.
As it take 30 minutes to mow this area, her mowing rate is:
= 192/30
= 6.4 square meters per minute.
The area of remaining lawn is:
= 13 x 32
= 416 square meters.
The minute to finish mowing is:
= 416 square meters / 6.4 square meters per minute.
= 65 minutes
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what is 9/11 - 9/14? Give your answer in its simplest form?
Answers
Answer: 0.175325
Step-by-step explanation:
Use a triple integral to find the volume of the solid enclosed by the paraboloids y = x^2 + z^2 and y = 32 - x^2 - z^2
Answers
The volume of the solid enclosed by the two paraboloids is approximately 1365.34π cubic units.
Here, we have to find the volume of the solid enclosed by the two paraboloids [tex]y = x^2 + z^2[/tex] and [tex]y = 32 - x^2 - z^2[/tex],
we need to set up a triple integral in cylindrical coordinates.
In cylindrical coordinates, we have:
x = r cos(θ)
z = r sin(θ)
y = y
The limits of integration for r, θ, and y will depend on the region of integration.
First, let's find the intersection points of the two paraboloids:
[tex]x^2 + z^2 = 32 - x^2 - z^2\\2x^2 + 2z^2 = 32\\x^2 + z^2 = 16[/tex]
This represents a circular region with radius r = 4 in the x-z plane.
Now, let's find the limits of integration for r, θ, and y:
For r:
Since the circular region has a radius of 4, the limits of r will be from 0 to 4.
0 ≤ r ≤ 4
For θ:
The intersection points form a complete circle in the x-z plane, so the limits of θ will be from 0 to 2π.
0 ≤ θ ≤ 2π
For y:
The lower paraboloid is given by [tex]y = x^2 + z^2[/tex], and the upper paraboloid is given by [tex]y = 32 - x^2 - z^2[/tex].
The limits of y will be from the lower paraboloid to the upper paraboloid.
[tex]x^2 + z^2 \leq y \leq 32 - x^2 - z^2[/tex]
Now, we can set up the triple integral to find the volume V:
V = ∫∫∫ (y) dy dθ dr
V = ∫∫∫ [tex](r^2 cos^2(\theta) + r^2 sin^2(\theta)) dy d\theta dr[/tex]
V = ∫∫∫ [tex](r^2) dy d\theta dr[/tex]
V = ∫∫ [tex](r^2) y|_{(x^2+z^2)}^{(32-x^2-z^2)} d\theta dr[/tex]
V = ∫∫ [tex](r^2) (32 - 2x^2 - 2z^2) d\theta dr[/tex]
V = ∫ [tex](32r^2 - 2r^2(x^2 + z^2)) d\theta dr[/tex]
V = ∫ [tex](32r^2 - 2r^2r^2) d\theta dr[/tex]
V = ∫ [tex](32r^2 - 2r^4) d\theta dr[/tex]
Now, integrate with respect to θ from 0 to 2π:
V = ∫(0 to 2π) ∫(0 to 4) [tex](32r^2 - 2r^4)[/tex] dr dθ
Now, integrate with respect to r from 0 to 4:
V = ∫(0 to 2π) [tex][(32/3)r^3 - (1/3)r^5] |_0 ^4[/tex] dθ
V = ∫(0 to 2π) [tex][(32/3)(4^3) - (1/3)(4^5)][/tex] dθ
V = ∫(0 to 2π) [(32/3)(64) - (1/3)(1024)] dθ
V = ∫(0 to 2π) [682.67] dθ
V = 682.67 * (2π) - 682.67 * (0)
V = 1365.34π cubic units
Therefore, the volume of the solid enclosed by the two paraboloids is approximately 1365.34π cubic units.
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can someone please help me, this is so complicated and i don’t understand how to do this :/
Answers
The slope of the line is 3.
Given are the coordinates of points on a line, we need to find the slope of the line,
Considering the points (2, 7) and (4, 13).
We know that the slope = y₂-y₁ / x₂-x₁
slope = 13-7 / 4-2 = 6/2
Slope = 3
Hence the slope of the line is 3.
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a farmer collects 78 artichokes and 90 courgettes. How many of the same cards can he form at most? If each artichoke is sold for $0.70 and each courgette for $0.40, how much does he get from the sale of 10 baskets
Answers
Step-by-step explanation:
To find how many of the same cards the farmer can form at most, we need to find the greatest common factor (GCF) of 78 and 90. One way to do this is to list the factors of each number and find the largest one they have in common:
78: 1, 2, 3, 6, 13, 26, 39, 78
90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
The largest factor they have in common is 6, so the farmer can form at most 6 baskets of the same size.
To find how much the farmer gets from the sale of 10 baskets, we need to first find the total number of artichokes and courgettes he has:
78 + 90 = 168
Next, we can find the total amount of money he gets from selling the artichokes and courgettes:
Money from artichokes = 78 × $0.70 = $54.60
Money from courgettes = 90 × $0.40 = $36.00
Total money = $54.60 + $36.00 = $90.60
If the farmer sells 10 baskets, then each basket would have 168/10 = 16.8 artichokes and courgettes. Since he can only form baskets of the same size, he would sell 6 baskets with 16 artichokes and 24 courgettes in each basket. The total amount of money he would get from selling these 6 baskets is:
Money from artichokes = 6 × 16 × $0.70 = $67.20
Money from courgettes = 6 × 24 × $0.40 = $57.60
Total money = $67.20 + $57.60 = $124.80
Therefore, the farmer would get $124.80 from the sale of 10 baskets.
Please what is A+5 divided by 2 =11
Answers
Answer:17
Step-by-step explanation:
you have to multiply both sides by 2 so A+5= 22 then subtract 5 over so A=17
add 2.75 to the product of 4.0 and 0.25 leave your answer in one decimal place
Answers
If we add 2.75 to the product of 4.0 and 0.25 then the value id 3.8
We have to find the value by adding 2.75 to the product of 4.0 and 0.25
The product of 4.0 and 0.25 is 1.0
4.0 × 0.25 = 1.0
Add 2.75 to the product of 4.0 and 0.25
2.75+1=3.75
Which is equal to 3.8
Therefore, if we add 2.75 to the product of 4.0 and 0.25 then the value id 3.8
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Find the volume of the right cone below. Round your answer to the nearest tenth if
necessary.
Answer:
units3
24
7
Submit Answer
Answers
Answer:
1231.5 units^2
Step-by-step explanation:
The formula for volume of a cone is
V = 1/3πr^2, where
V is the volume in cubic unitsr is the radius (find using circular base)and h is the height (line extending from the top of the cone to the circular base)We see from the diagram that the height is 24 cm and the radius is 7 cm. Now, we can plug these values in for h and r respectively in the volume formula:
V = 1/3π(7)^2(24)
V = 1/3π(49)(24)
V = 1/3π*1176
V = 392π
V = 1231.50432
V = 1231.5 cm^3
max flips k fair coins and sam flips k 1 fair coins. what is the probability that sam gets more heads than max?
Answers
Therefore, the probability that Sam gets more heads than Max is 0.5.
Let X be the number of heads Max gets and Y be the number of heads Sam gets. Both X and Y follow a binomial distribution with parameters k and 0.5.
We want to find P(Y>X), which can be expressed as:
P(Y>X) = P(Y-X>0)
Let Z = Y - X. Then Z follows a binomial distribution with parameters k and p = 0.5 - 0.5 = 0.
The mean of Z is E(Z) = E(Y-X) = E(Y) - E(X) = k/2 - k/2 = 0.
The variance of Z is Var(Z) = Var(Y-X) = Var(Y) + Var(X) = k/4 + k/4 = k/2.
Using the normal approximation to the binomial distribution, we can approximate Z as a normal distribution with mean 0 and variance k/2, for large enough values of k.
Therefore, P(Y>X) = P(Z>0) can be approximated using the standard normal distribution:
P(Z>0) = P((Z-0)/sqrt(k/2) > (0-0)/sqrt(k/2)) = P(Z/sqrt(k/2) > 0)
Using a standard normal table or calculator, we find that P(Z/sqrt(k/2) > 0) = 0.5.
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Find the missing side.
28°
N
15
cos(28°) =
?
Z
Remember: SOHCAHTOA
Enter
Answers
The missing side (the adjacent side) is approximately 13.257 units long.
To find the missing side, we can use the cosine function which relates the cosine of an angle to the adjacent side and the hypotenuse of a right triangle.
cos(28°) = adjacent side / hypotenuse
We are given the measure of the angle and the length of one side, so we can plug in those values and solve for the missing side.
cos(28°) = adjacent side / 15
adjacent side = 15 cos(28°)
Using a calculator to evaluate cos(28°) to four decimal places:
cos(28°) ≈ 0.8838
Substituting into the equation:
adjacent side ≈ 15 × 0.8838
adjacent side ≈ 13.257
Therefore, the missing side (the adjacent side) is approximately 13.257 units long.
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1. What is the area of the trapezoid? The diagram is not drawn to scale
O 36 cm^2
O 64 cm^2
O 128 cm^2
O 110 cm^2
11 cm
4cm
h
5 cm
Answers
Answer:
64cm^2 is the correct answer
you have $2.30 and would like to buy some rice flour. if a pound costs $4, how many ounces can you buy?
Answers
You have $2.30 and would like to buy some rice flour buy 9.2 ounces of rice flour with $2.30.
To arrive at this answer, we need to first convert the price per pound to price per ounce. There are 16 ounces in a pound, so the price per ounce is $4/16 = $0.25 per ounce.
Next, we divide the amount of money you have ($2.30) by the price per ounce ($0.25).
$2.30/$0.25 = 9.2 ounces.
Therefore, the conclusion is that you can buy 9.2 ounces of rice flour with $2.30.
Hi! I'm happy to help you with your question.
You can buy 9.2 ounces of rice flour.
1. First, we need to find out how many dollars you have per ounce of rice flour: $4 per pound / 16 ounces per pound = $0.25 per ounce.
2. Next, we'll determine how many ounces of rice flour you can buy with $2.30: $2.30 / $0.25 per ounce = 9.2 ounces.
With $2.30, you can buy 9.2 ounces of rice flour at the given price of $4 per pound.
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question 11 pleaseee help
Answers
The angle formed at the gas station is 130.3°. The correct option is (D).
Showing calculation for the angle of deviationRecall that, the area of a triangle is:
area = 1/2 * base * height
In this case, the base of the triangle is the distance between the two cars after 2 hours, and the height of the triangle is the distance from the gas station to the midpoint of the base.
Let's first find the distance traveled by each car in 2 hours:
distance traveled by car 1 = 45 mph * 2 hours = 90 miles
distance traveled by car 2 = 60 mph * 2 hours = 120 miles
The total distance between the two cars is the sum of their distances, which is:
total distance = 90 miles + 120 miles = 210 miles
The midpoint of the base is located at a distance of half the total distance from the gas station:
distance to midpoint = 1/2 * 210 miles = 105 miles
Now we can use the formula for the height of a triangle, which is:
height = 2 * area / base
Plugging in the values we have, we get:
height = 2 * 5190.713 mi²/ 210 miles = 123.824 mi
Finally, we can use the inverse tangent function to find the angle formed at the gas station, which is:
angle = tan⁻¹(height / distance to midpoint)
angle = tan⁻¹(123.824 mi / 105 miles)
angle = tan⁻¹(1.1793)
angle = 49.7°
However, this angle is the internal angle of the triangle formed by the two cars and the gas station. To find the angle formed at the gas station, we need to subtract this angle from 180°, which gives:
angle at gas station = 180° - 49.7° = 130.3°
Therefore, the answer is not one of the choices provided.
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The diagram shows a circle inside a rectangle.
2.5 cm
13.8 cm
Work out the area of the shaded region.
Give your answer correct to 3 significant figures.
(3 marks)
7.6 cm
Diagram NOT
accurately drawn
Answers
The area of the shaded region is given as follows:
85.245 cm².
How to obtain the area of the shaded region?The area of a rectangle is given by the multiplication of it's dimensions, hence it is given as follows:
Ar = 13.8 x 7.6
Ar = 104.88 cm².
The area of a circle of radius r is given by the equation presented as follows:
A = πr²
The radius for this problem is of r = 2.5 cm, hence the area is given as follows:
A = π x 2.5²
A = 19.635 cm²
Hence the area of the shaded region is given as follows:
104.88 - 19.635 = 85.245 cm².
Missing InformationThe diagram is given by the image presented at the end of the answer.
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birth weights of full-term babies in a certain area are normally distributed with mean 7.13 pounds and standard deviation 1.29 pounds. a newborn weighing 5.5 pounds or less is a low-weight baby. what is the probability that a randomly selected newborn is low-weight? group of answer choices about 21% of babies about 10.3% of babies about 89.7% of babies
Answers
Birth weights of full-term babies in a certain area are normally distributed with a mean of 7.13 pounds and a standard deviation of 1.29 pounds. a newborn weighing 5.5 pounds or less is a low-weight baby. There is a 10.3 % probability that a randomly selected newborn is low-weight. The correct answer is option B.
Given that the birth weights of full-term babies in a certain area are normally distributed with a mean of 7.13 pounds and a standard deviation of 1.29 pounds. A newborn weighing 5.5 pounds or less is a low-weight baby. We need to find the probability that a randomly selected newborn is low-weight.
Probability of newborn weighing 5.5 pounds or less = P(x ≤ 5.5)
We need to convert x to standard normal variable z.z = (x - μ) / σz = (5.5 - 7.13) / 1.29z = -1.26
Now, we need to find the area under the normal curve to the left of z = -1.26, using normal tables or calculators we get 0.1038A
Approximately, the probability that a randomly selected newborn is low-weight is 10.3%. Hence, option (B) is the correct answer.
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