We can use the Pythagorean theorem to find the distance between the game store and the skate park.
Let's call the distance between the game store and the skate park "d". Then we have:
d^2 = (distance between school and game store)^2 + (distance between school and skate park)^2
d^2 = 7.3^2 + 8.7^2
d^2 = 106.18
d ≈ 10.3
Therefore, the game store is approximately 10.3 kilometers from the skate park.
Bailey buys a car for $25,000. The car depreciates in value 18% per year. How much will the car be worth after 3 years? Round your answer to the nearest whole dollar amount.
A downward opening parabola with vertex (-5,2) and a vertical compression of 0. 5
The equation of the downward opening parabola with the given vertex and vertical compression is y = 0.5(x + 5)^2 + 2
The equation of a downward opening parabola with vertex (h, k) and vertical compression a is given by:
y = a(x - h)^2 + k
In this case, the vertex is (-5, 2) and the vertical compression is 0.5. Therefore, we have:
h = -5
k = 2
a = 0.5
Substituting these values into the equation above, we get:
y = 0.5(x + 5)^2 + 2
This is the equation of the downward opening parabola with the given vertex and vertical compression.
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Function f is defined by f(x)=2x+3. Function g is defined by g(y)=y^(2)-5. What is the value of (f(3)+g(-2)) ?
A. 0
B. 1
C. 2
D. 8
E. 10
Square oabc is drawn on a centimetre grid.o is (0,0) a is(3,0) b is(3,3) c is (0,3)write down how many invariants points there are on the perimeter of the square when oabc is translated by the vector (1 3)
There are 4 invariant points on the perimeter of the square when oabc is translated by the vector (1 3).
To find the invariant points on the perimeter of the square when oabc is translated by the vector (1 3), we need to apply this translation to each vertex of the square and see which ones remain on the square.
If we add the vector (1 3) to each vertex, we get:
o + (1 3) = (1 3)
a + (1 3) = (4 3)
b + (1 3) = (4 6)
c + (1 3) = (1 6)
Now we need to check which of these points are still on the square. We can see that points (1 3) and (4 3) are on two adjacent sides of the square, and points (1 6) and (4 6) are on the other two adjacent sides.
Therefore, there are 4 invariant points on the perimeter of the square when oabc is translated by the vector (1 3). These invariant points are the points where the sides of the original square intersect with the sides of the translated square.
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Find the directional derivative of f(x, y, z) = 23 – x²y at the point (3,-1, -2) in the direction of the vector v=(-1,-4,-4).
The directional derivative of f(x, y, z) = z³ – x²y at the point (3,-1, -2) in the direction of the vector v=(-1,-4,-4) is -234/√33.
The function is f(x, y, z) = z³ – x²y
We have to find directional derivative at the point (3, -1, -2)
In the direction vector v = (-1, -4, -4)
The gradient of the function is
∇f(x, y, z) = ∂f/∂x [tex]\hat{i}[/tex] + ∂f/∂y [tex]\hat{j}[/tex] + ∂f/∂z [tex]\hat{k}[/tex]
∇f(x, y, z) = ∂/∂x(z³ – x²y) [tex]\hat{i}[/tex] + ∂/∂y(z³ – x²y) [tex]\hat{j}[/tex] + ∂/∂z(z³ – x²y) [tex]\hat{k}[/tex]
∇f(x, y, z) = -2xy[tex]\hat{i}[/tex] - x²y[tex]\hat{j}[/tex] + 3z²[tex]\hat{k}[/tex]
At the point (3, -1, 4).
∇f(3, -1, 4) = -2(3)(-1)[tex]\hat{i}[/tex] - (3)²(-1)[tex]\hat{j}[/tex] + 3(4)²[tex]\hat{k}[/tex]
∇f(3, -1, 4) = 6[tex]\hat{i}[/tex] + 9[tex]\hat{j}[/tex] + 48[tex]\hat{k}[/tex]
The length of the vector is
|v| = √[(-1)² + (-4)² + (-4)²]
|v| = √[1 + 16 + 16]
|v| = √33
To normalize the vector we have
n = (-√33/33, -4√33/33, -4√33/33)
The directional derivative is
∇f(x, y, z) · n = (6, 9, 48) · (-√33/33, -4√33/33, -4√33/33)
∇f(x, y, z) · n = -6√33/33 - 36√33/33 - 192√33/33
∇f(x, y, z) · n = (-6 - 36 - 192)√33/33
∇f(x, y, z) · n = -234√33/33
∇f(x, y, z) · n = -234/√33
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PLEASE HELP!!
Line A has a slope of -1/3 and passes through the point (1, 10 1/3). Line B has a slope of 1/3 and passes through the point (-34, -2). Find the point where line A intersects like B.
The point where line A intersects line B is [2, 10].
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.At data point (1, 10 1/3) and a slope of -1/3, a linear equation for this line can be calculated or determined by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 10 1/3 = -1/3(x - 1)
y - 31/3 = -x/3 + 1/3
For Line B, we have:
y - y₁ = m(x - x₁)
y - (-2) = 1/3(x - (-34))
y + 2 = x/3 + 34/3
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Sound travels at an approximate speed of [tex]3.43(10^2)[/tex] m/s. How far will sound travel in 2 minutes?
Answer:41,160 meters in 2 minutes at the speed of 343 meters per second.
Step-by-step explanation:
The speed of sound varies depending on the medium it's traveling through, but assuming you meant the speed of sound in air at room temperature, it's approximately 343 meters per second.
To find out how far sound will travel in 2 minutes (120 seconds), we can simply multiply the speed of sound by the time:
Distance = Speed x Time
Distance = 343 m/s x 120 s
Distance = 41,160 meters
Therefore, sound will travel approximately 41,160 meters in 2 minutes at the speed of 343 meters per second.
Patrick and brooklyn are making decisions about their bank accounts. patrick wants to deposit $300 as a principle amount, with an interest of 6% compounded quarterly. brooklyn wants to deposit $300 as the principle amount, with an interest of 5% compounded monthly. explain which method results in more money after 2 years. show all work.
please give full explanation and work
Patrick's method of depositing $300 as the principle amount with an interest rate of 6% compounded quarterly results in more money after two years, with a final amount of $337.95.
To compare the two methods, we need to calculate the total amount of money each person will have after 2 years.
For Patrick:
The formula for compound interest is: A = P (1 + r/n)^(nt)
Where:
A = the total amount of money after t years
P = the principle amount
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
So for Patrick, we have:
A = 300 (1 + 0.06/4)^(4*2)
A = 300 (1.015)^8
A = 300*1.1265 = 337.95
After 2 years, Patrick will have $337.95.
For Brooklyn:
Using the same formula, we have:
A = 300 (1 + 0.05/12)^(12*2)
A = 300 (1.004167)^24
A = 300 * 1.10495 = 331.485
After 2 years, Brooklyn will have $331.485.
Therefore, Patrick's method of depositing $300 as the principle amount with an interest rate of 6% compounded quarterly results in more money after two years. Patrick will have $337.95, which is slightly more than Brooklyn with $331.485.
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Senior management of a consulting services firm is concerned about a growing decline in the firm's weekly number of billable hours. The firm expects each professional employee to spend at least 40 hours per week on work. In an effort to understand this problem better, management would like to estimate the standard deviation of the number of hours their employees spend on work-related activities in a typical week. Rather than reviewing the records of all the firm's full-time employees, the management randomly selected a sample of size 51 from the available frame. The sample mean and sample standard deviations were 48. 5 and 7. 5 hours, respectively. Construct a 88% confidence interval for the mean of the number of hours this firm's employees spend on work-related activities in a typical week. Place your LOWER limit, in hours, rounded to 1 decimal place, in the first blank. For example, 6. 7 would be a legitimate entry. ___ Place your UPPER limit, in hours, rounded to 1 decimal place, in the second blank. For example, 12. 3 would be a legitimate entry. ___
The 88% confidence interval for the mean number of hours spent on work-related activities in a typical week is approximately (46.9, 50.1).
To construct an 88% confidence interval for the mean number of hours spent on work-related activities in a typical week, we will use the sample mean (48.5 hours) and sample standard deviation (7.5 hours) from the sample of size 51.
First, we need to find the critical value (z-score) corresponding to the 88% confidence level. Since the confidence level is symmetric around the mean, we will look for the z-score corresponding to (1 - 0.88)/2 = 0.06 in each tail.
Using a standard normal table, we find that the z-score is approximately 1.56.
Now, we will calculate the margin of error using the formula:
Margin of error = z-score * (sample standard deviation / sqrt(sample size))
Margin of error = 1.56 * (7.5 / sqrt(51))
Margin of error ≈ 1.63
Next, we will calculate the confidence interval as follows:
Lower limit = sample mean - margin of error
Lower limit = 48.5 - 1.63
Lower limit ≈ 46.9
Upper limit = sample mean + margin of error
Upper limit = 48.5 + 1.63
Upper limit ≈ 50.1
So, the 88% confidence interval for the mean number of hours spent on work-related activities in a typical week is approximately (46.9, 50.1).
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4) what is the perimeter of the
trapezoid in simplest radical form?
helpp
The perimeter of the trapezoid in simplest radical form is 2(1 + 5√6).
The perimeter of the trapezoid = sum of sides
We know that 54 = 2 × 3 × 3 × 3
= 2 × 3³
24 = 2 × 2 × 2 × 3
= 2³ × 3
Perimeter = 2 + √54 + √54 + 2√24
= 2 + 2√54 + 2√24
= 2 + 2√(2 × 3³) + 2√(2³ × 3)
= 2 + 2 × 3√(2×3) + 2 × 2√(2 × 3)
= 2 + 6√(6) + 4√(6)
= 2 + 10√(6)
= 2(1 + 5√6)
Therefore, the perimeter of the trapezoid in simplest radical form is 2(1 + 5√6).
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Given question is incomplete, the complete question is below:
what is the perimeter of the trapezoid in simplest radical form?
Find the amount of force it takes to push jeff’s race car if the mass of the race car is 750 kg and the acceleration is 2. 5 startfraction m over s squared endfraction
the amount of force needed to push jeff’s race car is
newto
The amount of force required to push Jeff's race car is 1,875 Newtons (N).
How much force is required to push Jeff's race?The amount of force needed to push Jeff's race car is 1,875 Newtons (N), This problem provides us with the mass of Jeff's race car, which is 750 kg, and the acceleration it experiences, which is 2.5 m/s². We need to find the amount of force required to push the race car.
The formula to calculate force is:
Force = Mass x Acceleration
In this case, the mass of the race car is 750 kg and the acceleration is 2.5 m/s². We simply plug in these values into the formula to get:
Force = 750 kg x 2.5 m/s²
Simplifying the expression, we get:
Force = 1,875 N
Therefore, the amount of force required to push Jeff's race car is 1,875 Newtons (N).
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You roll a 6-sided die two times.
What is the probability of rolling a number greater than 1 and then rolling a number less than
3?
Answer:
Step-by-step explanation:
The possible outcomes of rolling a fair six-sided die are the numbers 1, 2, 3, 4, 5, and 6, each of which has an equal probability of $\frac{1}{6}$ of appearing.
The probability of rolling a number greater than 1 is $\frac{5}{6}$, since there are five out of six possible outcomes that satisfy this condition (namely, 2, 3, 4, 5, and 6).
The probability of rolling a number less than 3 is $\frac{2}{6}=\frac{1}{3}$, since there are two out of six possible outcomes that satisfy this condition (namely, 1 and 2).
To find the probability of both events happening (rolling a number greater than 1 and then rolling a number less than 3), we can multiply their respective probabilities:
$\frac{5}{6}\cdot\frac{1}{3}=\frac{5}{18}$
Therefore, the probability of rolling a number greater than 1 and then rolling a number less than 3 is $\boxed{\frac{5}{18}}$.
The snow globe below is formed by a hemisphere and a cylinder on a cylindrical
base. The dimensions are shown below. The base is slightly wider than the globe
with a diameter of 10cm and height of 1cm.
10 cm
4cm
3cm
1cm
Part D: The globes are ordered by the retail store in cases of 24. Design a rectangular
case to hold 24 globes packaged in individual boxes. What is the minimum
dimensions and volume of your case.
The minimum dimensions of the box will be; 7 cm × 6 cm × 6 cm
Since the dimension is described as the measurement of something in physical space such as length, width, or height.
Given that the there will be maximum dimension when the height of the cylinder and the radius of the hemisphere are aligned together.
Maximum height = 4 cm + 3 cm = 7 cm
Maximum diameter = 2 × 3 cm = 6 cm
Therefore, we can see that the minimum dimensions of the box are :
7 cm × 6 cm × 6 cm.
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What is the current ratio of length to width for us paper money
The current ratio of length to width for US paper money is approximately 2.61 to 6.14 inches. This means that US paper money is roughly rectangular in shape, with a length that is about 2.61 times greater than its width.
The current size of US paper money is standardized by the Bureau of Engraving and Printing (BEP). According to the BEP, the current size of a US paper bill is 2.61 inches wide and 6.14 inches long. This size has remained the same since the 1920s, although earlier bills were larger.
The rectangular shape of US paper money makes it easy to handle and store, and the standardized size ensures that it can be easily recognized and processed by vending machines, bank machines, and other automated devices.
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Sort each set of triangle measurements into the appropriate category for number of possible triangles. No Triangles One Triangle Many Triangles 5, 15", 160 45°, 45°, 90° 2.8. 10 7, 24, 25 30", 85°, 60° 5 of 5 Done
1) paul wants to deposit $7,300 into a one-year cd at a rate of 4.85%, compounded quarterly.
a) what his ending balance after the year?
b) how much interest did he earn?
c) what is his annual percentage yield?
hint: use the compounding interest formula
Using the compounding interest formula:
a) His ending balance after the year will be $7,658.91.
b) The amount of interest he will earn is $358.91.
c) His annual percentage yield is 4.9166%.
a) To calculate the ending balance after one year, we'll use the compound interest formula: A = P(1 + r/n)^(nt), where A is the ending balance, P is the principal ($7,300), r is the interest rate (4.85% or 0.0485), n is the number of compounding periods per year (4 for quarterly), and t is the number of years (1).
A = 7300(1 + 0.0485/4)^(4*1) = 7300(1.012125)⁴ = 7300*1.049166 = $7,658.91
b) To find the interest earned, subtract the principal from the ending balance: Interest = A - P
Interest = $7,658.91 - $7,300 = $358.91
c) To calculate the annual percentage yield (APY), we'll use the formula: APY = (1 + r/n)^(n) - 1
APY = (1 + 0.0485/4)⁴ - 1 = 1.049166 - 1 = 0.049166 or 4.9166%
Paul's ending balance after one year is $7,658.91, he earns $358.91 in interest, and his annual percentage yield is 4.9166%.
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MARK YOU THE BRAINLIEST !
Answer:
Angle C also measures 64°.
a consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. tube type a has mean brightness of 100 and standard deviation of 16, and tube type b has unknown mean brightness, but the standard deviation is assumed to be identical to that for type a. a random sample of tubes of each type is selected, and is computed. if equals or exceeds , the manufacturer would like to adopt type b for use. the observed difference is .
The probability that , Xb exceeds Xa , by 3.0 or more if ub and ua, are equal is 0.2537.
The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is stated as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more probable it is that the event will take place.
Probability is a way to gauge how likely something is to happen. Several things are difficult to forecast with absolute confidence. With it, we can only make predictions about the likelihood of an event happening, or how likely it is.
n1 = n2 = 25,
hypothesis,
standard error for difference,
[tex]\sqrt{\frac{16^2}{25} +\frac{16^2}{25} }[/tex]
=4.525
z =(3-0)/4.525
z=0.663
P(z ≥ 0.663) = 0.2537.
No, there is not strong evidence that [tex]\mu _B[/tex] is greater than [tex]\mu _A[/tex].
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Complete question;
A consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. Tube type A has mean brightness of 100 and standard deviation of 16, and tube type B has unknown mean brightness, but the standard deviation is assumed to be identical to that for type A. A random sample of n = 25 tubes of each type is selected, and X -X, is computed. If u, equals or exceeds u,, the manufacturer would like to adopt type B for use. The observed difference is X,X, - 3.0. a. What is the probability that , exceeds X, by 3.0 or more if ug and u, are equal? b. Is there strong evidence that ug is greater than u,?
Which set includes ONLY rational numbers that are also integers?
The set that includes ONLY rational numbers that are also integers is:
{-3, -2, -1, 0, 1, 2, 3, ...}
Which set includes ONLY rational numbers also integers?The set of rational numbers that are also integers is the set of numbers that can be expressed as a ratio of two integers where the denominator is 1. This means that the set includes numbers that are whole numbers, as well as their negatives.Learn more about integers
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The relative growth rate of a biomass at time t, R, is related to the concentration of a
substrate s at time t by the equation.
R(s) = cs / k+s
where c and k are positive constants.
What is the relative growth rate of the biomass if there is no substrate present?
If there is no substrate present, the concentration of s would be 0. The relative growth rate of biomass at time t, R, is related to the concentration of a substrate s at time t by the equation R(s) = cs / (k+s), where c and k are positive constants.
To find the relative growth rate of the biomass if there is no substrate present, we need to set the concentration of the substrate, s, to 0. Using the given equation, we can substitute 0 for s:
R(0) = c(0) / k + 0
R(0) = 0 / k
R(0) = 0
Therefore, the relative growth rate of the biomass would be 0 if there is no substrate present.
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The food company is now designing soup boxes. The largest box of soup will be a dilation of the smallest box using a scale factor of 2. The Smallest box hold 8 fl oz or about 15 cubic inches of soup find a set of dimensions for the largest box? round your answer to the nearest tenth if necessary
The largest box of soup will hold about 120 ounces or 221 cubic inches of soup.
Since the scale factor is 2, the volume of the largest box will be 2^3 = 8 times the volume of the smallest box. Therefore, the volume of the largest box will be 8 x 15 cubic inches = 120 cubic inches. To find the dimensions of the largest box, we need to find the cube root of 120 cubic inches, which is approximately 5.87 inches.
Since the smallest box has no shape restrictions, we can assume that the largest box will also have a rectangular shape. Therefore, a set of dimensions for the largest box could be 5.87 inches x 5.87 inches x 5.87 inches, or rounded to the nearest tenth, 5.9 inches x 5.9 inches x 5.9 inches.
This would result in a volume of approximately 221 cubic inches, which is about 120 ounces of soup.
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Approximate, in square meters, the area of a circle with diameter equal to
5/6
meters. Leave your answer in fraction form. (Use
22/7 to approximate. )
The area of the circle is (121/144)π square meters.
We know that the formula for the area of a circle is A = πr², where r is the radius of the circle. However, we are given the diameter of the circle, which is 5/6 meters.
The diameter of a circle is twice the radius, so we can find the radius by dividing the diameter by 2:
radius = (5/6) / 2 = 5/12 meters.
Now that we have the radius, we can use the formula for the area of a circle:
A = πr² = π(5/12)².
To approximate this using 22/7, we first simplify (5/12)²:
(5/12)² = 25/144.
Substituting this value into the formula, we get:
A = π(25/144) = (25/144)π.
Therefore, the area of the circle is (25/144)π square meters. To get an approximation, we can use 22/7 approximate π:
A ≈ (25/144) × (22/7) = 121/144 square meters.
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A triangle has an area of 52 in², what would the area be if the base was one half as long and the height was twice as long?
If the base was one half as long and the height was twice as long, then the area of the triangle will be 52 in².
To find the area of a triangle, we use the formula: area = (base × height) / 2. Given that the original triangle has an area of 52 square inches, we can represent this as: 52 = (base × height) / 2.
Now, let's consider the new triangle, where the base is half as long and the height is twice as long. This can be represented as base' = base / 2 and height' = height × 2.
Using the formula for the area of the new triangle, we have: area' = (base' × height') / 2 = ((base / 2) × (height × 2)) / 2.
By simplifying the equation, we see that the factors of 2 cancel out, leaving us with: area' = (base × height) / 2.
As we know that the area of the original triangle is 52 square inches, we can conclude that the area of the new triangle will also be 52 square inches. This is because the changes to the base and height essentially cancel each other out, resulting in the same overall area.
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An aluminum can is to be constructed to contain 2500 cm of liquid. Letr and h be the radius of the base and the height of the can respectively. a) Express h in terms of r. (If needed you can enter y aspi.) h = b) Express the surface area of the can in terms of r. Surface area = C) Approximate the value of r that will minimize the amount of required material (i.e. the value of that will minimize the surface area). What is the corresponding value of h? TE h=
a) We can use the formula for the volume of a cylinder to relate the given liquid volume to the dimensions of the can: πr^2h = 2500, Solving for h, we get: h = 2500/(πr^2)
b) The surface area of the can consists of the area of the circular top and bottom, as well as the area of the cylindrical side. The area of the top and bottom is 2πr^2 each, and the area of the side is 2πrh. Therefore, the total surface area is: Surface area = 2πr^2 + 2πrh
Substituting the expression for h in terms of r that we found in part (a), we get:
Surface area = 2πr^2 + 2πr(2500/(πr^2))
Simplifying, we get:
Surface area = 2πr^2 + 5000/r
c) To minimize the surface area, we need to find the value of r that makes the derivative of the surface area with respect to r equal to zero. So we differentiate the expression we found in part (b) with respect to r: d(Surface area)/dr = 4πr - 5000/r^2
Setting this equal to zero and solving for r, we get:
4πr = 5000/r^2
r^3 = 1250/π
r ≈ 6.17 (rounded to two decimal places)
Substituting this value of r into the expression we found for h in part (a), we get: h ≈ 10.55 (rounded to two decimal places)
Therefore, the aluminum can should have a radius of approximately 6.17 cm and a height of approximately 10.55 cm in order to minimize the surface area and conserve material.
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Lucy is running a test on her car engine that requires her car to be moving. The tolerance for the variation in her car’s speed, in miles/hour, while running the test is given by the inequality |x − 60| ≤ 3. Assume x is the actual speed of the car at any time during the test
The car's speed must be within the range of 57 to 63 miles per hour to stay within the specified tolerance.
To determine the range of speeds Lucy's car can be moving within the given tolerance, we can analyze the inequality |x - 60| ≤ 3, where x is the actual speed of the car in miles per hour.
Step 1: Break the absolute value inequality into two separate inequalities:
(x - 60) ≤ 3 and -(x - 60) ≤ 3
Step 2: Solve each inequality:
For (x - 60) ≤ 3:
x ≤ 60 + 3
x ≤ 63
For -(x - 60) ≤ 3:
-x + 60 ≤ 3
-x ≤ -57
x ≥ 57
Step 3: Combine the solutions to get the range of allowable speeds:
57 ≤ x ≤ 63
So, when Lucy is running the test on her car engine, the car's speed must be within the range of 57 to 63 miles per hour to stay within the specified tolerance.
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The rent for an apartment was $6,600 per year in 2012. If the rent increased at a rate of 4% each year thereafter, use an exponential equation to find the rent of the apartment in 2017. (Write your answer in dollars, such as $XX. XX)
The rent for the apartment using exponential equation in 2017 was $8,029.91.
To find the rent of the apartment in 2017, we will use an exponential equation. An exponential equation is a mathematical expression where a variable is raised to a power, often used to model growth or decay. In this case, we will model the growth of the rent over time.
1. Identify the initial rent, the growth rate, and the number of years that have passed since 2012.
Initial rent (A0): $6,600
Growth rate (r): 4% = 0.04
Number of years (t): 2017 - 2012 = 5
2. Write the exponential equation for the rent increase:
At = A0 * (1 + r)^t
3. Plug in the given values and calculate the rent in 2017:
At = $6,600 * (1 + 0.04)^5
4. Calculate the rent:
At = $6,600 * (1.04)^5
At = $6,600 * 1.2166529
At = $8,029.91
The rent for the apartment in 2017 was $8,029.91. This was calculated using an exponential equation, which allowed us to account for the 4% annual increase in rent over the 5 years since 2012.
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Find the general solution to y"’+ 4y" + 40y' = 0. In your answer, use C1, C2 and C3 to denote arbitrary constants and x the independent variable.
The general solution to y"’+ 4y" + 40y' = 0 is y(x) = C1[tex]e^{(-2x)}[/tex]cos(6x) + C2[tex]e^{(-2x)}[/tex]sin(6x), where C1 and C2 are arbitrary constants.
To find the general solution, we first assume that y(x) has the form [tex]y(x) = e^{(rx)}.[/tex]
Substituting this into the differential equation, we get the characteristic equation r³ + 4r² + 40r = 0.
Factoring out r, we get r(r² + 4r + 40) = 0. The quadratic factor has no real roots, so we can write r = 0, -2 ± 6i.
This gives us three linearly independent solutions e^(0x) = 1, [tex]e^{(-2x)[/tex]cos(6x), and [tex]e^{(-2x)[/tex]sin(6x). Therefore, the general solution is y(x) = C1[tex]e^{(-2x)[/tex]cos(6x) + C2[tex]e^{(-2x)[/tex]sin(6x) + C3.
Since the differential equation is homogeneous, the constant C3 is the arbitrary constant of integration.
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A)Irene purchased some earrings that regularly cost $55 for a friend’s birthday. Irene used a "20% Off" coupon.
How much did Irene pay for the earrings?
Show your work. Highlight your answer.
B)Irene’s friend did not like the gift so she tried to return the earrings. She did not have the receipt, so the store would only give her store credit for 50% of the purchase price.
How much credit did Irene’s friend receive?
Show your work. Highlight your answer.
C)What is the percent change from what Irene paid and what her friend returned it for?
Show your work. Highlight your answer
A) Irene pays $44 for the earrings.
B) Irene’s friend received $22 as credit.
C) Percent change from what Irene paid and what her friend returned it for is 50%
A) Cost of earing = $55
Discount coupon = 20%
Total cost Irene pay = 55 - (20% of 55)
Total cost Irene pay = 55 - ( 55 × 20/100)
Total cost Irene pay = 55 - 11
Total cost Irene pay = 44
B) Credit given by store = 50%
Credit received = 50% of 44
Credit received = 44 × 50/100
Credit received = 22
C) Percent change = [tex]\frac{final - initial }{initial}[/tex] × 100
Percent change = [tex]\frac{44-22}{44}[/tex] × 100
Percent change = 50%
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at state college last term, 50 of the students in a physics course earned a's, 75 earned b's, 114 got c's, 98 were issued d's, and 50 failed the course. if this grade distribution was graphed on pie chart, how many degrees would be used to indicate the b region? round your answer to the nearest whole degree, but do not include a degree symbol with your response.
The angle in degrees used to indicate the b region is 70.
The total number of students= Sum of the number of students with different grades and the failed ones.
= 50+75+114+98+50
= 387
Now,
The number of students in b region, that is, those who got b's
=75 (given)
We know that,
The sum of all angles due to different grades in the pie chart = 360 degrees.
So the distribution of degrees to b region in the pie chart will be in proportion to the number of students in b region out of total students
Let x degrees be used to indicate the "b" region.
∴ x/360=75/387 (because of the same proportion)
⇒x=75/387×360
⇒x=69.76≅70
Hence, the angle in degrees used to indicate the b region is 70.
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if f(x) - x ^ 2 + 1 6(x) = 3x and fg(x) = gf(x) find the value of x
The value of x is [tex]\sqrt{\frac{2}{6} }[/tex]
What is a function?A function can be defined as a law or expression showing the relationship between two variables.
From the information given, we have that;
f(x) = x ^ 2 + 1
g(x) = 3x
To determine the composite function, substitute the value of the function inside the bracket and the value of x in the other function, we have;
fg(x) = (3x²) + 1
expand the bracket
fg(x) = 9x² + 1
Then,
gf(x) = 3(x² + 1)
expand the bracket
gf(x) = 3x² + 3
Equate the functions, we have;
9x² + 1 = 3x² + 3
collect the like terms
6x² = 2
Divide the value
x = [tex]\sqrt{\frac{2}{6} }[/tex]
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