Answers
The area of the shaded portion of the given shape is: 114 yd²
What is the area of the shaded region?The formula for the area of a rectangle is expressed as:
A = L * W
Where:
L is Length
W is Width
Now, to find the area of the shaded part, we will find the area of the bigger rectangle and subtract the area of the unshaded rectangle from it.
Thus:
Area of shaded part = (20 * 12) - (14 * 9)
Area of shaded part = 240 - 126
Area of shaded part = 114 yd²
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Related Questions
Given the problem
ut = uxx, 0 < x < 2, t<0
u(x, 0) = 4x(2 - x) 0 < x < 2
u(0,t) = u(2, t) = 0, t > 0 using the energy method show that the integral 2∫0 u^2 (x, t) dx is a decreasing function of t.
Answers
By using the energy method, we get integral is a decreasing function of time t.
To use the energy method, we first multiply the given PDE by u and integrate over the domain:
∫[0,2]∫[0,t] u*ut dxdt = ∫[0,2]∫[0,t] u*uxx dxdt
Using integration by parts and the given boundary conditions, we can simplify this expression to:
d/dt (∫[0,2] u^2 dx) = -2∫[0,2] u^2 dx
This shows that the integral ∫[0,2] u^2 dx is a decreasing function of t.
Therefore, the energy of the system is decreasing over time, indicating that the solution is stable.
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pleasehelp due today
Answers
Answer:
11
Step-by-step explanation:
Volume of right cone = (1/3) · π · r² · h
V = 968π
h = 24 units
Let's solve
968π = (1/3) · π · r² · 24
2904π = π · r² · 24
121π = π · r²
121 = r²
r = 11
So, the radius is 11 units.
Suppose the incubation period for certain types of cold viruses are normally distributed with a population standard deviation of 8 hours. Use Excel to calculate the minimum sample size needed to be 99% confident that the sample mean is within 4 hours of the true population mean.Be sure to round up to the nearest integer.
Answers
The minimum sample size needed to be 99% confident that the sample mean is within 4 hours of the true population mean is 27.
To calculate the minimum sample size needed to be 99% confident that the sample mean is within 4 hours of the true population mean with a population standard deviation of 8 hours, follow these steps:
1. Identify the desired confidence level: In this case, it is 99%.
2. Find the corresponding Z-score for the confidence level: For a 99% confidence level, the Z-score is approximately 2.576.
3. Identify the population standard deviation: In this case, it is 8 hours.
4. Identify the margin of error: In this case, it is 4 hours.
5. Use the following formula to calculate the sample size:
Sample size (n) = (Z-score^2 * population standard deviation^2) / margin of error^2
Plugging in the values, we get:
n = (2.576^2 * 8^2) / 4^2
n = (6.635776 * 64) / 16
n = 26.543104
Since we need to round up to the nearest integer, the minimum sample size needed is 27.
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30 60 90 special right triangle
Answers
The values of x and y are 16 and 16√3 respectively
What are special angles in trigonometry?The special angles on the unit circle refer to the angles that have corresponding coordinates which can be solved with the Pythagorean Theorem.
These special angles includes: 30°,45°, and 60°
sin 30 = 1/2, cos 60 = 1/2 , cos 30 = √3/2 , sin60 = √3/2 e.t.c
therefore,
sin30 = x/32
1/2 = x/32
2x = 32
x = 32/2 = 16
cos 30 = adj/hyp
√3/2 = y/32
2y = 32√3
y = 32√3/2
y = 16√3
therefore the values of x and y are 16 and 16√3 respectively.
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How many people will 5 pitchers serve if 1/8 pitcher served one person
Answers
Using proportion, we can see that 5 pitchers will serve 40 people if 1/8 pitcher served one person.
Given that,
1/8 pitcher served one person.
Let x be the number of people that the 5 pitchers served.
We can find the value using the proportional method.
Using the proportional concept, the ratio of the number of pitchers served to the number of people will be proportional.
So,
(1/8) / 1 = 5 / x
1/8 = 5/x
Cross multiplying,
x = 40
Hence 5 pitchers will serve 40 people.
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HELP PLEASE I NEED THE ANSWER ASAP
Answers
It says that the landscaping company uses 3 1/2 tons the first month and then it says the next month uses the same amount on each of the five prodjects so for every one project in the second month they use 3 1/2 tons.
X³ + 3x² + 3x + 1 ÷ x - 1/2
Answers
We have found that the remainder when x³ + 3x² + 3x + 1 is divided by x + 1 is 0.
How do we describe the Remainder theorem?The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a).
Given:
f(x) = x³ + 3x² + 3x + 1
We first calculate the remainder of polynomial f(x) when divided by (x + 1).
We apply the Remainder theorem, the remainder of f(x) when divided by (x - r) is f(r).
we then determine the value of f(-1).
Substituting value of x = -1 is given polynomial f(x).
f(x) = x³ + 3x² + 3x + 1
f(-1) = (-1)³ + 3(-1)² + 3(*-1) +1
f(-1) = 0
In conclusion, the remainder when x³ + 3x² + 3x + 1 is divided by x + 1 is 0.
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A country can use all its resources to produce Product A and Product B. If you know the opportunity cost of
producing Product A in terms of Product B, how can you quickly determine the cost of Product B in terms of
product A? Explain in one to two sentences, using an example.
Answers
You can take the reciprocal of the opportunity cost of producing Product A in terms of Product B to determine the cost of producing Product B in terms of Product A,
To determine the cost of producing Product B in terms of Product A, you can take the reciprocal of the opportunity cost of producing Product A in terms of Product B.
If the opportunity cost of producing 1 unit of Product A is 2 units of Product B, then the cost of producing 1 unit of Product B would be 1/2 unit of Product A.
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BRAINLIEST FOR THE BEST ANSWER!!!! HELP ASAP
find the surface area of this triangular prism.
the surface area of the shape is 1. ____ 2. ____ inches.
1. 24, 48, 96, 108
2. square, cubic
Answers
Answer:
The surface area of the shape is 1. 96 2. square inches.
1. 96
2. square
Step-by-step explanation:
We can see that the given right triangular prism is composed of the following regular 2D shapes whose areas add up to the total surface area of the prism
A vertical rectangle of size 6 in. x 2 in.Area of this rectangle = 6 x 2 = 12 square inchesA horizontal rectangle of size 8 in. x 2in.
Area of this rectangle = 8 x 2 = 16 square inchesA rectangle along the hypotenuse of the prism with dimensions 2 in. x 10 in.
Area of this rectangle = 2 x 10 = 20 square inchesTwo triangles on the sides of the prism.
Each triangle has a height of 6 in. and a base of 8 in.
Area of a triangle = 1/2 (bh) where b = base and h = height
Area of each triangle = 1/2 x 8 x 6 = 24 square inches
Area of both triangles = 2 x 24 = 48 square inches
Total surface area of triangular prism = 12 + 16 + 20 + 48
= 96 square inches
Answer:
96 square inches
Step-by-step explanation:
To figure out how much surface area a right triangular prism has, you gotta break it down into a few 2D shapes. There's a tall rectangle that's 6 inches wide and 2 inches tall, which is 12 square inches. Then there's a wide rectangle that's 8 inches wide and 2 inches tall, which is 16 square inches. There's also a rectangle on the diagonal part of the prism that's 2 inches wide and 10 inches tall, which is 20 square inches. And don't forget the two triangles on the sides! Each triangle is 6 inches tall and 8 inches wide, which is 24 square inches each, for a total of 48 square inches. Add all those areas up and bam, you've got the total surface area of the triangular prism - which in this case is 96 square inches.
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An elementary teacher wants to know if the school has a higher proportion of left-handed students than the usual proportion of 0.10. The teacher surveys a random sample of 50 students, and finds that 7 are left-handed. 1) What is the sample proportion ? O 0.14 O 0.10 07 2) What is the hypothesized proportion po? O 0.14 O 0.5 O 0.10 3) What is the sample size n? O 50 O 7 4) What is the test statistic z? O 0.943 0 -0.815
Answers
1) The sample proportion is 0.14 (7 left-handed students out of 50 total students surveyed).
2) The hypothesized proportion po is 0.10 (the usual proportion of left-handed students).
3) The sample size n is 50 (the number of students surveyed).
4) The test statistic z is 1.32.
1) The sample proportion is calculated by dividing the number of left-handed students by the total number of students surveyed. In this case, 7 left-handed students out of 50 gives a sample proportion of 7/50 = 0.14.
2) The hypothesized proportion (p₀) is the usual proportion of left-handed students, which is given as 0.10.
3) The sample size (n) is the total number of students surveyed, which is 50.
4) The test statistic (z) can be calculated using the formula: z = (sample proportion - hypothesized proportion) / sqrt((hypothesized proportion * (1 - hypothesized proportion)) / sample size). In this case, z = (0.14 - 0.10) / sqrt((0.10 * (1 - 0.10)) / 50) = 0.04 / sqrt(0.09 / 50) ≈ 0.943.
To calculate the test statistic z, we use the formula:
z = (sample proportion - hypothesized proportion) / standard error
The standard error is calculated as:
standard error = sqrt((po * (1-po)) / n)
Plugging in the values, we get:
standard error = sqrt((0.10 * (1-0.10)) / 50) = 0.0499
Then,
z = (0.14 - 0.10) / 0.0499 = 1.32
Since the calculated z-value of 1.32 is greater than the critical value of 1.645 (using a significance level of 0.05 for a two-tailed test), we can conclude that there is not enough evidence to reject the null hypothesis that the proportion of left-handed students at the school is the same as the usual proportion of 0.10.
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Show that the function f(x)= 1 3x3−2x2 7x has no relative extreme points. Relative extreme points exist when____f'(x)=0 or f''(x)=0___. In this case, because _f'(x) or f''(x)____=_____. ____has no x-int, has no y-int, has multiple x-int, has multiple y-int____ the function f(x)=2/3x^3-4x^2+10x has no relative extreme points
Answers
The f'(x) has two x-intercepts, but f''(x) is always positive, indicating that f(x) has no relative extrema. This means that the function is either always increasing or always decreasing, and there are no maximum or minimum points.
The function f(x) =
[tex](1/3)x^3 - (2/7)x^2 - 1x[/tex]
has no relative extreme points. To find the relative extreme points of a function, we need to find the critical points where either the derivative f'(x) is equal to zero or the second derivative f''(x) is equal to zero.
Taking the derivative of f(x), we get f'(x) = x^2 - (4/7)x - 1. Setting f'(x) equal to zero and solving for x, we get x =
[tex](2 ± \sqrt{} (30))/7[/tex]
Upon further analysis of the second derivative f''(x) = 2x - (4/7), we see that it is always positive for all values of x.
There are no relative extreme points as the function f(x) does not have any points where the slope is zero and the curvature changes from positive to negative or vice versa.
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Sharon stands on the top of a cliff 90 m high. The angle of elevation from Sharon to a flying kittiwake is 15°. The angle of depression from Sharon to a yacht on the sea is 19º.
Given that the kittiwake is flying
directly above the yacht, find the
distance between the yacht and the kittiwake.
Answers
The distance between the yacht and the kittiwake. is 160 m
How to find the distance between the yacht and the kittiwakeThe horizontal distance between Sharon and the yacht
tan 19 = 90 / distance between Sharon and the yacht
distance between Sharon and the yacht = 90 / tan 19
distance between Sharon and the yacht = 261.38 m
The horizontal distance between the Sharon and the kittiwake
tan 15 = distance between the Sharon and the kittiwake / 261.38
distance between the Sharon and the kittiwake = 261.38 x tan 15
distance between the Sharon and the kittiwake = 70.04 m
distance between the yacht and the kittiwake
= 90 + 70.04
= 160.04
= 160 m
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(PART A)The general form of a circle is given as
x^2+y^2+4x-12y+4=0.
(a) What are the coordinates of the center of the circle?
(b) What is the length of the radius of the circle?
Answer:
(PART B)
A 10-foot ladder placed on level ground leans against the side of a house. The ladder reaches a point that is 9.2 feet up on the side of the house.
(a) What is the measure of the angle formed by the ladder and the level ground? Round your answer to the nearest degree. Show your work.
(b) The Occupational Safety and Health Administration (OSHA) sets standards for a variety of occupations to help prevent accidents and other safety hazards. OSHA’s standard for the angle formed by a ladder and level ground is 75°. The same 10-foot long ladder is placed against the building according to OSHA’s safety standard.
What is the distance between the foot of the ladder and the foot of the building? Round your answer to the nearest tenth. Show your work.
Answer:
Answers
The distance between the foot of the ladder and the foot of the building is 2.6 ft
How to solvePart 1) The general form of a circle is given as x²+y² +4x - 12y + 4 = 0.
(a)What are the coordinates of the center of the circle?
(b)What is the length of the radius of the circle?
x²+y² +4x - 12y + 4 = 0
Group terms that contain the same variable, and move the constant to the opposite side of the equation
(x²+4x)+(y²- 12y)=-4
Complete the square twice. Remember to balance the equation by adding the same constants to each side
(x²+4x+4)+(y²- 12y+36)=-4+4+36
Rewrite as perfect squares
(x+2)²+(y-6)²=36--------> (x+2)²+(y-6)²=6²
center (-2,6)
radius 6
the answer Part a) is
the center is the point (-2,6)
the answer Part b) is
the radius is 6
Part 2)
see the picture attached N 1 to better understand the problem
we know that
sin ∅=opposite side angle ∅/hypotenuse
opposite side angle ∅=9.2 ft
hypotenuse=10 ft
so
sin ∅=9.2/10-----> 0.92
∅=arc sin (0.92)------> ∅=66.93°-----> ∅=67°
the answer Part a) is
67°
Part b)
see the picture attached N 2 to better understand the problem
cos 75=adjacent side angle 75/hypotenuse
adjacent side angle 75=AC
hypotenuse=10 ft
so
cos 75=AC/10---------> AC=10*cos 75----> AC=2.59 ft----> AC=2.6 ft
the answer Part B) is
The distance between the foot of the ladder and the foot of the building is 2.6 ft
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A bridge connecting two cities separated by a lake has a length of 4.042 mi.
Use the table of facts to find the length of the bridge in yards.
Round your answer to the nearest tenth.
Answers
The length of the lake of 4.042 miles in yards is 7113.92 yards
How long is the length in yards?From the question, we have the following parameters that can be used in our computation:
Lake has a length of 4.042 mi.
This means that
Length = 4.042 miles
From the table of values:
To convert inches to feet, we multiply the length value by 1760
So, we have
Length = 4.042 * 1760 yards
Evaluate
Length = 7113.92 yards
Hence, the length is 7113.92 yards
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2x^2-13x+7=x^2 to the nearest tenth
Answers
The answer for your math question is x= 12.44, x = 0.56
what is the answer to this question -5(x+2)=5
Answers
The solution to the equation -5(x + 2) = 5 is x = -3.
What is the solution to the given equation?Given the equation in the question:
-5( x + 2 ) = 5
First, we distribute the -5 to the expression inside the parenthesis:
-5×x + 2×-5= 5
-5x - 10 = 5
Next, let's isolate the variable x by adding 10 to both sides:
-5x - 10 + 10 = 5 + 10
-5x = 5 + 10
Simplifying the left side:
-5x = 5 + 10
-5x = 15
Finally, we can solve for x by dividing both sides by -5:
-5x / -5 = 15 / -5
x = -3
Therefore, the value of x is -3.
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Deon rented a truck for one day. There was a base fee of $20.95, and there was an additional charge of 74 cents for each mile driven. Deon had to pay $221.49 when he returned the truck. For how many miles did he drive the truck?
Answers
Answer:
Deon drove 270 miles.
Let x be the number of miles driven.
The total cost is 20.95 + 0.74x = 221.49
Subtracting 20.95 from both sides gives 0.74x = 200.54
Dividing both sides by 0.74 gives x = 270 miles
So the answer is 270
Answer:
271 miles
Step-by-step explanation:
The equation to find the total cost is
C = 20.95 + .74 m where m is the number of miles
221.49 = 20.95 +.74m
Subtract 20.95 from each side
221.49 -20.95 = 20.95-20.95 +.74m
200.54 = .74m
Divide each side by .74
200.74/.74 = m
271 = m
Help me and thank you.
Answers
The volume of the given cube is determined as (q cm)³.
What is the volume of the cube?The volume of a cube is calculated from the cube its edge length.
Mathematically, the formula for the volume of a cube is calculated by applying the following formula.
V = L x L x L = L³
where;
L is the edge length of the cubeThe volume of the given cube is calculated as follows;
V = q cm x q cm x q cm
V = (q cm)³
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help asap please
A dog is tied to a wooden stake in a backyard. His leash is 3 meters long and he runs around in circles pulling the leash as far as it can go. How much area does the dog have to run around in? Use 3.14 for pi.
Answers
The area the dog have to run around in is 28.26 square meters
How much area does the dog have to run around in?From the question, we have the following parameters that can be used in our computation:
His leash is 3 meters long and he runs around in circles
This means that
Radius, r = 3 meters
The area is calculated as
Area = 3.14r^2
So, we have
Area = 3.14 * 3^2
Evaluate
Area = 28.26
Hence, the area is 28.26 square meters
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A windowpane is 15 inches by 8 inches. What is the distance between opposite corners of the windowpane?
Answers
The distance between the opposite corners of the windowpane would be 17 inches.
How to find the distance ?If the windowpane is divided diagonally, we see that the distance between the opposite corners can be be the hypotenuse of a right angle triangle.
This allows us to use the Pythagorean theorem to find that distance between opposite sides. The distance is:
d ² = 15 ² + 8 ²
d ² = 225 + 64
d ² = 289
d = √ 289
d = 17 inches
In conclusion, the distance between the opposite corners is 17 inches.
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An integer divided by 7 gives a result-3.What is that integer
Answers
Answer:
- 21
Step-by-step explanation:
let the integer be n , then
[tex]\frac{n}{7}[/tex] = - 3 ( multiply both sides by 7 to clear the fraction )
n = 7 × - 3 = - 21
that is the integer is - 21
From the problem, we know that:
x ÷ 7 = -3
To solve for "x", we can multiply both sides of the equation by 7:
x = -3 x 7
x = -21
Therefore, the integer we're looking for is -21.
4. The image below is a cube. Edge BC is along the x-axis. Edge BA is along the y-axis. Edge BF
is along the z-axis. What are the coordinates of point G?
E
F
3
2
Bo 1
D
Answers
In order to precisely locate a point, you must identify its location relative to a coordinate system.
How to find the coordinates of a pointThis system assigns numerical values to points in space and is thus equipped to uniquely pinpoint any position.
Firstly, ascertain the distance from the point to the coordinate system's origin.
Applying this knowledge to a Cartesian-style grid, consider both the horizontal (x-coordinate) as well as vertical (y-coordinate) distances between the specified point and the origin.
Finally, render the coordinates of the target point in the correct style of notation.
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a question i need to do on my homework
Answers
Answer:
Step-by-step explanation:
1. Make an equation. To stay balanced the sum of everything on the left must equal the value on the right.
2 + x + 2 + x + 2 + x + 2 + x + 2 + x = 11
2. Add all like terms. We have 2+2+2+2+2 = 10 and x+x+x+x+x=5x so the equation simplifies to
10 + 5x = 11
3. Substract 10 from both sides
10 - 10 + 5x = 11 - 10
0 + 5x = 1
5x = 1
4. Divide the number being multiplied by x from both sides
5x/5 = 1/5
x = 1/5 or x = 0.2
how much time do americans spend eating or drinking? suppose for a random sample of 1001 americans, the mean time eating or drinking per day is 1.22 hours with a sample standard deviation of 0.65 hours. (a) construct and interpret a 99% confidence interval for the mean amount of time americans spend eating or drinking per day. (b) suppose you want to conduct your own survey. using the sample standard deviation above, how large of a sample is required to estimate the mean time americans spend eating or drinking per day within 15 minutes of the true mean and with 95% confidence?
Answers
a. we are 99% confident that the true population mean time Americans spend eating or drinking per day falls between 1.166 and 1.274 hours.
b. There will be 70 sample is required to estimate the mean time Americans spend eating or drinking per day within 15 minutes of the true mean and with 95% confidence
(a) To construct a 99% confidence interval for the mean time Americans spend eating or drinking per day, we can use the formula:
CI = x ± z*(σ/√n)
where x is the sample mean, σ is the population standard deviation (which is unknown, so we use the sample standard deviation), n is the sample size, and z* is the critical value for a 99% confidence interval (which we can find using a table or calculator).
Plugging in the values given, we get:
CI = 1.22 ± 2.58*(0.65/√1001) ≈ 1.22 ± 0.054
So the 99% confidence interval for the mean time Americans spend eating or drinking per day is (1.166, 1.274) hours.
We can interpret this interval as saying that we are 99% confident that the true population mean time Americans spend eating or drinking per day falls between 1.166 and 1.274 hours.
(b) To find the sample size required to estimate the mean time Americans spend eating or drinking per day within 15 minutes of the true mean with 95% confidence, we can use the formula:
n = (z*σ/E)^2
where E is the margin of error (which is 15 minutes = 0.25 hours), z* is the critical value for a 95% confidence interval (which is 1.96), and σ is the sample standard deviation (which is 0.65).
Plugging in the values given, we get:
n = (1.96*0.65/0.25)^2 ≈ 69.88
So we need a sample size of at least 70 to estimate the mean time Americans spend eating or drinking per day within 15 minutes of the true mean with 95% confidence.
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Let CD be a line segment of length 6. A point P is chosen at random on CD. What is the probability that the distance from P to C is smaller than the square of the distance from P to D? Hint: If we think of C as having coordinate 0 and D as having coordinate 6, and P as having coordinate, then the condition is equivalent to the inequality < (6 − x)²
Answers
The probability that the distance from P to C is smaller than the square of the distance from P to D is 1/3.
Given a line segment CD of length 6.
A point P is chosen at random on CD.
Let C(0, 0) and D (6, 0).
Any point in between C and D will be of the form (x, 0).
So let P (x, 0).
Then using distance formula,
CP = √x² = x
PD = √(6 - x)² = 6 - x
CP < (PD)²
x < (6 - x)²
x < 36 - 12x + x²
x² - 13x + 36 > 0
(x - 9)(x - 4) > 0
x - 9 > 0 and x - 4 > 0
x > 9 and x > 4
x > 9 is not possible.
Hence x > 4.
Possible lengths are 5 and 6.
Probability = 2/6 = 1/3
Hence the required probability is 1/3.
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Aiman is 4 years ago older than his younger brother. The product of Aiman and his younger brother's ages is equal to their father's age. The father is 48 years old and Aiman's younger brother is (p) years old. Write a quadratic equation in terms of (p).
Answers
p^2 + 4p - 48 = 0
Let's first express Aiman's age in terms of his younger brother's age. If Aiman is four years older than his younger brother, then we can write:
Aiman's age = younger brother's age + 4
Let p be the age of Aiman's younger brother. Then Aiman's age is p + 4.
The product of Aiman and his younger brother's ages is equal to their father's age, which is given as 48. So we can write:
(p + 4) * p = 48
Expanding the left-hand side:
p^2 + 4p = 48
Subtracting 48 from both sides:
p^2 + 4p - 48 = 0
This is a quadratic equation in terms of p.
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A train travels 75 feet in 44 second. At the same speed, how many feet will it travel in 5 seconds?
Answers
If a train travels 75 feet in 44 seconds. At the same speed, it travels 8.52 feet in 5 seconds
To find out how many feet the train will travel in 5 seconds at the same speed, first, we need to determine the speed of the train.
The train travels 75 feet in 44 seconds. To find the speed, we'll divide the distance traveled (75 feet) by the time taken (44 seconds):
Speed = Distance / Time
Speed = 75 feet / 44 seconds
Now, we can calculate the distance the train will travel in 5 seconds at the same speed:
Distance = Speed × Time
Distance = (75 feet / 44 seconds) × 5 seconds
The "seconds" unit cancels out, and we're left with:
Distance = (75 feet / 44) × 5
Now, we can calculate the distance:
Distance ≈ 8.52 feet
So, at the same speed, the train will travel approximately 8.52 feet in 5 seconds.
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Suppose your parents have 2 options to purchase a plot of land on which they plan to build a barn.
Option 1: They can purchase the land for $30,000 cash.
Option 2: They can purchase the land with $7,500 down, and then pay $2,500 semi-annually for the next 10 years,
at an interest rate of 5%.
Calculate the present value for both options, and tell which will save them the most money.
Option 1 will save your parents the most money.
Option 2 will save your parents the most money.
It is not possible to determine which option will save the most money because the question does not state how large the
barn will be.
The options both cost the same, so neither one will save them money.
Answers
Answer:
PV = $30,000; this saves the mostPV = $46,473 — the higher-cost optionStep-by-step explanation:
You want the present value and the lower-cost choice for two payment plans:
$30,000 cash$7500 down and $2500 semi-annually for 10 years at 5%Present valueThe present value of 20 semiannual payments of $2500 discounted at the rate of 5% can be found by a financial calculator to be $38,973. Together with the $7500 down payment, the present value of Option 2 is ...
Option 2 = $7500 +38,973 = $46,473
The present value of $30,000 cash is $30,000.
ComparisonOption 1 has a present value of $30,000.
Option 2 has a present value of $46,473.
Option 1 will save your parents the most money.
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Additional comment
The total cash outlay for option 2 is $7500 + 20×2500 = $57,500. For this option to be the same cost as option 1, the account would need to earn interest at the rate of 18.4%.
There are various ways to estimate the interest earned. One of them is to compute half the value of simple interest on the interval. That is, the interest could be estimated as (1/2)(5%/yr)(10 yr) = 25%. This suggests the PV would be about 1/1.25 times the sum of payments, or 40000. That's close enough to the actual value of 39000 to tell you that Option 1 is the better choice.
A flashlight battery is guaranteed to last for 40 hours. Test indicates that the length of life of these batteries is normally distributed with mean 50 and variance 16. What percentage of the batteries fail to meet the guarantee?
Answers
To find the percentage of batteries that fail to meet the guarantee, we need to calculate the probability that the battery lasts less than 40 hours. Since we know that the length of life of these batteries is normally distributed with mean 50 and variance 16, we can use the z-score formula:
z = (x - μ) / σ
where x is the value we want to find the probability for (in this case, x = 40), μ is the mean (μ = 50), and σ is the standard deviation (σ = sqrt(16) = 4).
So, we have:
z = (40 - 50) / 4 = -2.5
Looking up the probability for a z-score of -2.5 in a standard normal distribution table, we find that the probability is 0.0062, or 0.62%.
Therefore, approximately 0.62% of the batteries fail to meet the guarantee.
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The following population data of a basic design of a product are given as:
1. Base product
average length - 90 cm
with a standard deviation of the length - 7 cm
2. A modifications was made to this product and a sample of 12 unit was collected. The sample is shown in the table to the right:
3. Test at a=0.01 whether there is difference between standard deviations (+/-) of this product's length between the base and the modified product?
a) What is/are the critical value(s)? b) What is/are the test statistic(s)? c) Was there a difference? Yes or No
Answers
a) The critical value for a two-tailed test with a significance level of 0.01 and 11 degrees of freedom is 3.11 (found using a t-distribution table).
b) The test statistic for comparing two standard deviations is the F-statistic. The formula for calculating it is [tex]F = \frac{s1^2}{s2^2}[/tex], where s1 is the sample standard deviation of the first group (base product), s2 is the sample standard deviation of the second group (modified product), and the larger standard deviation is always in the numerator. Using the sample data given, we find:
s1 = 7 cm (from the base product)
s2 = 6.5 cm (from the modified product)
[tex]F = \frac{(7)^{2} }{(6.5)^{2} } = 1.223[/tex]
c) To determine if there is a difference between the standard deviations, we compare the calculated F-statistic to the critical value we found in part a. Since our calculated F-value (1.223) is less than the critical value (3.11), we fail to reject the null hypothesis. Therefore, we conclude that there is not enough evidence to suggest that there is a significant difference between the standard deviations of the two products.
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