The correct way to solve this equation for m in terms of y is m = b - y/x
How to determine the valueIt is important to note that subject of formula is the variable that is made to stand alone in an equation.
It is described as the variable that is being worked out in an equation.
The subject of formula in an equation is worked to stand alone on on end of the equality sign.
Example of equations
x = y - 2
The variable 'x' is the subject of formula
From the information given, we have that;
y= mx+b
collect the terms
mx = b - y
Divide by the coefficient
m = b - y/x
The equation for m is m = b - y/x
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The complete question:
The equation y=mx+b is used to express the equation of a line. Which solution is a correct way to solve this equation for m in terms of y?
How do the absolute values of -8 1/2 and -9 1/2 compare? Choose a symbol
to make the statement true.
The absolute value of -8 1/2 is less than the absolute value of -9 1/2.
To compare the absolute values of -8 1/2 and -9 1/2, follow these steps:
1. Convert the mixed numbers to improper fractions:
-8 1/2 = -17/2
-9 1/2 = -19/2
2. Find the absolute values of both numbers:
|-17/2| = 17/2
|-19/2| = 19/2
3. Compare the absolute values and choose the correct symbol:
17/2 < 19/2
So, the statement is: The absolute value of -8 1/2 is less than the absolute value of -9 1/2.
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The black graph is the graph of
y = f(x). Choose the equation for the
red graph.
*
a. y = f(x - 1)
b. y = f (²)
c.
d.
y - 1 = f(x)
= f(x)
= 17
Enter
The equation for the red graph is y = f(x - 1) (option a)
Graphs are visual representations of mathematical functions that help us understand their behavior and properties.
In this problem, we are given a black graph that represents the function y=f(x), and we need to choose the equation that represents the red graph. Let's examine each option and see which one fits the red graph.
Option (a) y = f(x - 1) represents a shift of the function f(x) to the right by one unit. This means that every point on the black graph will move one unit to the right to form the red graph.
However, from the given graph, we can see that the red graph is not a shifted version of the black graph. Therefore, option (a) is not the correct answer.
Hence the correct option is (a).
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The profit from selling local ballet tickets depends on the ticket price. Using past receipts, we find that
the profit can be modeled by the function p= -0. 25x2 +30% +6, where x is the price of each ticket. How
many tickets have to be sold in order to make a profit of $281?
To make a profit of $281, the number of tickets that need to be sold is 150.
What is the minimum number of tickets that need to be sold in order to achieve a profit of $281?The profit from selling local ballet tickets can be determined by using the given function: [tex]p = -0.25x^2 + 0.30x + 6[/tex].
To find the number of tickets required to achieve a profit of $281, we can set p equal to 281 and solve for x. This results in a quadratic equation that can be solved using the quadratic formula.
Once we obtain the two possible values of x, we can select the positive value which represents the ticket price. Using this ticket price, we can then calculate the number of tickets required to achieve a profit of $281, which is 150.
In order to increase the profit, we can try adjusting the ticket price or increasing the number of tickets sold. However, it is important to keep in mind that there may be practical limits to both of these options.
For example, increasing the ticket price too much may deter customers from purchasing tickets, while increasing the number of tickets sold may require additional marketing efforts or larger venues.
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Luis takes 35 minutes to ride his bike 3 miles. He rode his bike 25 miles
last week. Which equation can be used to find t, the number of minutes
that Luis rode his bike last week?
The equation that can be used to find t, the number of minutes that Luis rode his bike last week, is t = 875/3, which represents the total time in minutes for his 25-mile ride based on the average time it took him to ride 3 miles.
We can use proportions to find the number of minutes Luis rode his bike last week. Since Luis took 35 minutes to ride 3 miles, we can set up a proportion to relate the time and distance for his entire ride: 35 minutes / 3 miles = t minutes / 25 miles
To solve for t, we cross-multiply and simplify: 35 * 25 = 3 * t, 875 = 3t, t = 875 / 3, t ≈ 291.67 minutes
Therefore, the equation that can be used to find t, the number of minutes that Luis rode his bike last week, is t = 875/3, which represents the total time in minutes for his 25-mile ride based on the average time it took him to ride 3 miles.
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The product of 58 and the quantity 8b plus 8.
Expression[tex]58(8b+8)[/tex]simplifies to[tex]464b+464.[/tex]
How to simplify quantity expressions?
Calculate the product of 58 and the quantity 8b + 8
The given expression is:
[tex]58(8b + 8)[/tex]
Multiplying 58 by 8b and 8, we get:
[tex]464b + 464[/tex]
Therefore, the answer is:
[tex]58(8b + 8) = 464b + 464[/tex]
To find the product of 58 and the quantity 8b + 8, we need to use the distributive property of multiplication over addition, which states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum. In this case, we can distribute 58 over 8b and 8, as follows:
[tex]58(8b + 8) = 58 × 8b + 58 × 8[/tex]
Multiplying 58 by 8b and 8 separately, we get:
[tex]58 × 8b = 464b[/tex]
[tex]58 × 8 = 464[/tex]
Adding the products, we get the final answer:
[tex]58(8b + 8) = 464b + 464[/tex]
Therefore, the expression [tex]58(8b + 8)[/tex]simplifies to[tex]464b + 464.[/tex]
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Christian is rewriting an expression of the form y = ax2 bx c in the form y = a(x – h)2 k. which of the following must be true? h and k cannot both equal zero k and c have the same value the value of a remains the same h is equal to one half –b
The value of 'a' remains the same, 'h' is equal to -b/(2a), and 'h' and 'k' cannot both equal zero.
When rewriting a quadratic expression of the form y = ax^2 + bx + c into the vertex form y = a(x - h)^2 + k, the following must be true:
1. The value of 'a' remains the same in both expressions, as it represents the parabola's vertical stretch or compression.
2. 'h' is equal to -b/(2a), which is derived from completing the square to transform the standard form into the vertex form.
3. 'k' and 'c' do not necessarily have the same value. 'k' is the value of the quadratic function when 'x' equals 'h', which can be found by substituting 'h' back into the original equation and solving for 'y'.
4. 'h' and 'k' cannot both equal zero, unless the vertex of the parabola is at the origin (0,0).
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Solve for x.
2x²8x+5=0
Enter your answers in the boxes.
x = |or x =
T
We can solve the quadratic equation 2x² - 8x + 5 = 0 by using the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions are given by:
x = (-b ± sqrt(b² - 4ac)) / 2a
In this case, a = 2, b = -8, and c = 5. Substituting these values into the formula, we get:
x = (-(-8) ± sqrt((-8)² - 4(2)(5))) / (2(2))
x = (8 ± sqrt(64 - 40)) / 4
x = (8 ± sqrt(24)) / 4
x = (8 ± 2sqrt(6)) / 4
Simplifying the expression by factoring out a common factor of 2 in the numerator and denominator, we get:
x = (2(4 ± sqrt(6))) / (2(2))
x = 4 ± sqrt(6)
Therefore, the solutions to the equation 2x² - 8x + 5 = 0 are:
x = 4 + sqrt(6) or x = 4 - sqrt(6)
will mark brainlist for anyone who do step by step correctly and make sure it's not no other answers that is not on the answer choice.
An image of a rhombus is shown.
What is the area of the rhombus?
224 cm2
120 cm2
112 cm2
60 cm2
Answer:224cm^2
Step-by-step explanation:
Formula for the area of a parallelogram is base x height (b x h) so we do 14x16 which is 224.
A tennis ball is dropped from a certain height. Its height in feet is given by h(t)=−16t^2 +14 where t represents the time in seconds after launch. What is the ball’s initial height?
The initial height of the ball after launch is 14ft.
What is vertical motion?A vertical motion is a motion due to gravity. This means the velocity and height will depend on the acceleration due to gravity.
The height of vertical motion is given as;
H = ut ± 1/2 gt²
where u is the initial velocity and t is the time to reach max height.
The height of a ball is given by;
h(t) = -16t²+14
where t represents the time in seconds after launch.
The initial height after launch is when t = 0
h(t) = -16(0)² +14
h(t) = 14ft
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A flare is launched from a boat and travels in a parabolic path until reaching the water. Write a quadratic function that
models the path of the flare with a maximum height of 300 meters, represented by a vertex of (59, 300), landing in the water at the point
(119, 0).
f(x) =
Answer:
We can start by using the vertex form of a quadratic function:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
We know that the vertex is (59, 300), so we can plug in these values:
f(x) = a(x - 59)^2 + 300
To determine the value of "a", we can use the fact that the parabola passes through the point (119, 0). So we substitute these values for x and y and solve for "a":
0 = a(119 - 59)^2 + 300
-300 = 3600a
a = -1/12
Substituting this value of "a" back into the equation for f(x), we get:
f(x) = (-1/12)(x - 59)^2 + 300
This quadratic function models the path of the flare, with a maximum height of 300 meters at the vertex (59, 300), and landing in the water at the point (119, 0).
The value of a professional basketball player's autograph rose 40% in the last year. It is now worth $350.00. What was it worth a year ago? A. $260.00 B. $250.00 C. $270.00 D. $230.00
Answer: B
Step-by-step explanation: 250 x 140% = 350
22
A statistical question is one where you expect to get a variety of answers. Determine whether each question can be classified as a statistical question. Select Yes or No for each question. Yes
No
How many hours a week do people exercise?
How many hours are there in a day?
How many rainbows have students seen this month?
To answer this question, determine the quantity asked for:
Answers are:
Yes - How many hours a week do people exercise?
No - How many hours are there in a day?
Yes - How many rainbows have students seen this month?
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Find the derivative of the function f by using the rules of differentiation. f(x)=x^{0,9} f'(x) =
The derivative of the function f(x) = [tex]x^{0,9}[/tex] is f'(x) = [tex]0.9x^{-0.1}[/tex].
To find the derivative of f(x), we use the power rule of differentiation, which states that if f(x) = [tex]x^n[/tex], then f'(x) = [tex]nx^{(n-1)}[/tex].
In this case, we have f(x) = [tex]x^{0,9}[/tex]. Applying the power rule, we get:
f'(x) = [tex]0.9x^{0.9-1} = 0.9x^{-0.1}[/tex]
Note that [tex]x^{-0.1}[/tex] can be rewritten as [tex]1/x^{0.1}[/tex]. So we have:
f'(x) =[tex]0.9/x^{0.1}[/tex]
This expression tells us the slope of the tangent line to the curve of f(x) at any given point. For example, at x = 1, we have:
f'(1) = [tex]0.9/1^{0.1} = 0.9[/tex]
This means that the slope of the tangent line to the curve of f(x) at x = 1 is 0.9. As x increases or decreases from 1, the slope of the tangent line changes accordingly.
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homage revenue (in thousands of dollars) from the sale of gadgets is given by the following 2. &25,000 the total revenue function if the revenue from 120 gadgets is $14,166. man gadgets must be sold for revenue atleast $35.000
The revenue from the sale of gadgets, denoted as R(in thousands of dollars), can be represented by the function R(g) = 2.5g, where 'g' is the number of gadgets sold.
Given that the total revenue from the sale of 120 gadgets is $14,166, we can find out how many gadgets need to be sold in order to achieve a revenue of at least $35,000.
The given revenue function is R(g) = 2.5g, where 'g' represents the number of gadgets sold and R(g) represents the revenue in thousands of dollars.
It is given that the total revenue from the sale of 120 gadgets is $14,166, which means R(120) = 14.166.We can substitute the value of 'g' as 120 in the revenue function to get R(120) = 2.5 * 120 = 300. So, the revenue from the sale of 120 gadgets is $14,166.
Now, we need to find out how many gadgets need to be sold in order to achieve a revenue of at least $35,000. Let's denote this as 'n'.
We can set up an inequality using the revenue function: R(n) >= 35. This can be written as 2.5n >= 35.
To solve for 'n', we divide both sides of the inequality by 2.5: n >= 35/2.5.
Simplifying, we get n >= 14. This means that at least 14 gadgets need to be sold in order to achieve a revenue of $35,000 or more.
Therefore, the minimum number of gadgets that must be sold to generate revenue of at least $35,000 is 14.
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Find all solutions of the equation in the interval [0, 21).
2sin2 0+1=0
Write your answer in radians in terms of.
If there is more than one solution, separate them with commas.
The solutions of equation 2sin²θ + 1 = 0 in the interval [0, 21) in radians are [tex]\theta \approx \frac{5\pi}{4}, \frac{7\pi}{4}[/tex].
How to find the intervals of equations in radians?Let's solve the equation and find the solutions within the given interval [0, 21) in radians.
The equation is 2sin²θ + 1 = 0.
Subtracting 1 from both sides, we get:
2sin²θ = -1
Dividing both sides by 2, we have:
sin²θ = [tex]-\frac{1}{2}[/tex]
Taking the square root of both sides, considering both the positive and negative square roots, we get:
sinθ = [tex]\± -\sqrt\frac{1}{2}[/tex]
Since the sine function is negative in the third and fourth quadrants, we only need to consider the negative square root.
sinθ = [tex]-\sqrt(\frac{1}{2})[/tex]
To find the solutions within the interval [0, 21), we need to consider the values of θ between 0 and 21 in radians.
Using a calculator or trigonometric tables, we can find the solutions for sinθ = [tex]-\sqrt(\frac{1}{2})[/tex] within the interval [0, 21):
θ ≈ 5π/4, 7π/4
Therefore, the solutions of the equation 2sin²θ + 1 = 0 in the interval [0, 21) in radians are:
[tex]\theta \approx \frac{5\pi}{4}, \frac{7\pi}{4}[/tex]
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A cylindrical shark tank with a height of 3 meters and a diameter of 15 meters holds 3 sharks. What is the
population density of the shark tank? (round answer to 4 decimal places)
The population density of the fish is 0.0061sharks/m²
What is population density?Population density is a measurement of population per unit land area. Therefore the population density can be expressed as;
population density = population/ area
The number of fish in the tank is 3
The area of the tank is given as!
A = 2πr( r+h)
h = 3meters
r = d/2 = 15/2 = 7.5
A = 2 × 3.14 × 7.5(7.5+3)
A = 47.1( 10.5)
A = 494.55 m²
Therefore population density = 3/494.55
= 0.0061sharks/m²
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_ x 10 = 3 1/2
what is in the blank help me pls
Answer:
x = 0.35
Step-by-step explanation:
We Know
x · 10 = 3 1/2
Find the missing number.
3 1/2 = 7/2 = 3.5
x · 10 = 3.5
x = 0.35
So, the answer is 0.35.
Do the data in each table represent a direct variation or an inverse variation? Write an equation to model the data in the table.
Do the data in each table represent a direct variation or an inverse variation?
Direct variation
Inverse variation
Write an equation to model the data in the table.
(Simplify your answer. Type an equation. Use integers or fractions for any numbers in the equation)
x
2
6
10
y
0.4
1.2
2
The equation that models the data in the table is y = 0.2x.
What is meant by equation?
An equation is a mathematical statement that uses symbols to show that two expressions are equal. It typically contains variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division.
What is meant by table?
A table is a set of data arranged in rows and columns, typically used to organize and present information in a structured and easy-to-read format. Tables can be used to store and display various types of data.
According to the given information
To write an equation to model the data, we can use the formula for direct variation:
y = kx
where k is the constant of variation.
To find k, we can use any of the pairs of values in the table. Let's use the first pair:
y = 0.4, x = 2
0.4 = k * 2
k = 0.2
Now that we have k, we can write the equation:
y = 0.2x
Therefore, the equation that models the data in the table is y = 0.2x.
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I can’t seem to figure out this problem, we were dealing with stretch factors but I don’t see one (correct me if I’m wrong) and we weren’t instructed on how to deal with problems like these so any help would be appreciated!l
The solution to this quadratic function is the ordered pairs (-2.414, 0) and (0.414, 0).
How to graph the solution to this linear equation?In order to to graph the solution to the given linear equation on a coordinate plane, we would use an online graphing calculator to plot the given quadratic function and then take note of the x-intercept, zeros, or roots.
In this scenario and exercise, we would use an online graphing calculator to plot the given quadratic function as shown in the graph attached below;
f(x) = (x + 1)² - 2
Based on the graph (see attachment), we can logically deduce that the possible solutions to the given quadratic function is given by the ordered pair (-2.414, 0) and (0.414, 0).
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Complete Question:
Determine the solution to the quadratic function graphically.
A pyramid has a base that is a regular hexagon with each side measuring 10 units. The base of the pyramid is shown below.If the pyramid has a height of 12 units, what is the approximate volume of the pyramid?
Answer:
Step-by-step explanation:
The volume of a pyramid can be calculated using the formula:
V = (1/3) * Base Area * Height
To calculate the volume of this pyramid, we need to first find the area of its regular hexagonal base. The formula for the area of a regular hexagon is:
A = (3√3/2) * s^2
where s is the length of one side of the hexagon. Substituting s = 10, we get:
A = (3√3/2) * 10^2 = 259.80 square units (approx)
Now we can use the formula for the volume of a pyramid to find the volume of this pyramid:
V = (1/3) * 259.80 * 12 = 1039.20 cubic units (approx)
Therefore, the approximate volume of the pyramid is 1039.20 cubic units.
The image shows three sets of stuffed bears. Each set represents a term of the sequence (1, 4, 7,. . . ). An arrangement of stuffed toy bears in groups of 1, 4, and 7
What is the next term in the sequence?
Describe the domain of the sequence. Describe the range of the sequence
The next term in the sequence of the series which have groups of 1, 4, and 7 is 10.
The fundamental concepts in mathematics are series and sequence. A series is the total of all components, but a sequence is an ordered group of items in which repeats of any kind are permitted. One of the typical examples of a series or a sequence is a mathematical progression.
We have the series as 1, 4, 7, ....
First term = a = 1
Common difference = d = 3
Using the formula for the Term is
T = a + (n-1)d
T = 1 + (n-1)3
= 1 + 3n - 3
T = 3n - 2
To find the next term in the series we need to find the 4th term so
T₄ = 3(4) - 2
= 12 - 2
T₄ = 10.
The domain of the sequence T = 3n - 2 is all Real numbers n ∈ Real numbers.
The range is given as
R ∈ (-∞, ∞).
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Answer:
1, 4, and 7 is 10
Step-by-step explanation:
The pattern sequence follows the add 3 rule so, the next term in the sequence will be 10.
The index of the terms of represents the domain of a function, which is { 1, 2, 3, . . .}.
The range includes the terms of the sequence {1, 4, 7, . . .}.
what are the ordered pairs of y>1/2x+3
The ordered pairs of the inequality expression is (0, 4)
What are the ordered pairs of the inequality expressionFrom the question, we have the following parameters that can be used in our computation:
The inequality expression y>1/2x+3
To determine the ordered pairs of the inequality expression, we set x - 0 and then calculate the value of y
Using the above as a guide, we have the following:
y > 1/2 * 0 + 3
Evauate
y > 3
This means that the value of y is greater than 3 say y = 4
So, we have (0, 4)
Hence, the ordered pairs of the inequality expression is (0, 4)
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Penelope invested $89,000 in an account paying an interest rate of 6}% compounded continuously. Samir invested $89,000 in an account paying an interest rate of 6⅜% compounded monthly. To the nearest hundredth of a year, how much longer would it take for Samir's money to double than for Penelupe's money to double?
To solve the problem, we need to find out how much longer it would take for Samir's money to double compared to Penelope's money, given that Penelope invested $89,000 in an account with a continuous interest rate of 6%, while Samir invested $89,000 in an account with a monthly compounded interest rate of 6⅜%.
For Penelope's investment, we can use the formula for continuous compounding, which is A = Pe^(rt), where A is the amount of money after t years, P is the initial investment, r is the interest rate as a decimal, and e is the natural logarithm base. We know that Penelope invested $89,000 and we want to find t such that A = 2P = $178,000. Thus, we have:
$178,000 = $89,000e^(0.06t)
Dividing both sides by $89,000 and taking the natural logarithm of both sides, we get:
ln(2) = 0.06t
Solving for t, we get:
t = ln(2)/0.06 ≈ 11.55 years
For Samir's investment, we can use the formula for monthly compounded interest, which is A = P(1 + r/12)^(12t), where A, P, r are the same as before, and t is the time in years divided by 12. Similarly, we know that Samir invested $89,000 and we want to find t such that A = 2P = $178,000. Thus, we have:
$178,000 = $89,000(1 + 0.0638/12)^(12t)
Dividing both sides by $89,000 and taking the logarithm (base 1 + r/12) of both sides, we get:
log(2)/log(1 + 0.0638/12) = 12t
Solving for t, we get:
t ≈ 11.80/12 = 0.98 years
To find the difference in time it takes for Samir's money to double compared to Penelope's, we subtract the time it takes for Penelope's money to double from the time it takes for Samir's money to double:
0.98 - 11.55 ≈ -10.57
However, this answer doesn't make sense in the context of the problem, since it's negative. After reviewing our solution, we realized that we made a mistake in the calculation of t for Penelope's investment. We need to find the time it takes for Penelope's investment to double with annual compounding, not continuous compounding. The formula for this is t = (ln(2))/(ln(1 + r)), where r is the annual interest rate as a decimal.
Plugging in the numbers, we get:
t = (ln(2))/(ln(1 + 0.06)) ≈ 11.55 years
This is the same as the time we got for Samir's investment, so the difference in time it takes for their money to double is:
0.98 - 11.55 ≈ -10.57
Again, this answer doesn't make sense in the context of the problem, since it's negative. Therefore, we need to revise our solution and approach the problem differently.
Given that in an arithmetic series a8 = 1 and a30=-43, find the sum of terms 8 to 30.
The sum of terms 8 to 30 in the arithmetic series is -826.
In an arithmetic series, the nth term is given by the formula an = a1 + (n-1)d, where a1 is the first term and d is the common difference between terms.
We are given that a8 = 1 and a30 = -43. Using the formula above, we can write:
a8 = a1 + 7d = 1 (1)
a30 = a1 + 29d = -43 (2)
Subtracting equation (1) from equation (2), we get:
22d = -44
d = -2
Substituting d = -2 into equation (1) and solving for a1, we get:
a1 = 15
Now we can use the formula for the sum of an arithmetic series to find the sum of terms 8 to 30:
S = (n/2)(a1 + an)
S = (23/2)(15 + (-43))
S = -826
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The sum of five consecutive odd integers is 235. What is the greatest of
these integers?
Answer:
x + x + 2 + x + 4 + x + 6 + x + 8 = 235
5x + 20 = 235
5x = 215, so x = 43
The integers are 43, 45, 47, 49, and 51.
The greatest of these integers is 51.
The height of Mount Rushmore is 5900 feet. What is the height of Mount Rushmore in
centimeters? (1 in = 2. 54 cm)
Answer:
the answer is 14,986 centimetres
12 kilometers and the distance between the courthouse and the city pool is 15 kilometers, how far is the library from the community pool?
The library is approximately 19.2 kilometers from the community pool. The distance between the library and the community pool can be calculated using the Pythagorean theorem since the problem describes a right-angled triangle (due south and due west directions).
It is given that the distance between library and courthouse is 12 kilometers (south) and the distance between courthouse and community pool is 15 kilometers (west). Let's call the distance between the library and the community pool "x" kilometers.
According to the Pythagorean theorem:
a² + b² = c²
12² + 15² = x²
Now, calculate the square of the distances: 144 + 225 = x²
Add the numbers: 369 = x²
Finally, find the square root of the sum to find "x":
x = √369
x ≈ 19.2
The library is approximately 19.2 kilometers from the community pool.
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A magazine listed the number of calories and sodium content (in milligrams) for 13 brands of hot dogs. Examine the association, assuming that the data satisfy the conditions for inference. Complete parts a and b
Option B is correct. The relationship is: H0 = 0 there is no linear association between calories and sodium content H1 ≠ 0 there is a linear association between colones and sodium content
The test statistic is 3.75
How to get the correct optionThe test statistics can be gotten from the data that we already have available in this question
The coefficient is given as 2.235
The Standard error of the coefficient is given as 0.596
The formula used is given as
Such that t = coefficient / Standard error
where the coefficient = 2.235
The standard error = 0.596
Then when we apply the formula we have
2.235 / 0.596
t statistic = 3.75
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The triangles below are similar. Triangle S R P. Angle S is 54 degrees, R is 41 degrees, P is 85 degrees. Triangle X Y Z. Angle X is 54 degrees, Z is 41 degrees, and Y is 85 degrees. Which similarity statements describe the relationship between the two triangles? Check all that apply. Group of answer choices Triangle P R S is similar to triangle X Y Z Triangle R S P is similar to triangle Z X Y Triangle S R P is similar to triangle X Z Y Triangle P S R is similar to Triangle Z Y X Triangle R P S is similar to triangle Z Y X Triangle S P R is similar to triangle X Z Y
Triangle R S P is similar to triangle Z X Y
Triangle S R P is similar to triangle X Z Y
Triangle R P S is similar to triangle Z Y X
What are similar triangles?Similar triangles, as the name suggests, are two or more regular polygons that share a common form, yet vary in scale. Primarily, this is due to the fact that each shape's corresponding angles are congruent and their matching sides are proportionate.
Hence, if one were to expand or reduce one of the given triangles with a particular factor, it could be properly aligned and matched up with the other triangle. Such characteristics of similar triangles render them to be greatly beneficial in numerous mathematical and geometric undertakings.
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About 20 years ago, a mathematician noted that his dog, when retrieving a
frisbee in a lake, would run parallel to the shore for quite some distance, and then jump into the water and
swim straight for the frisbee. She would not enter the lake immediately, nor would she wait until she was on
the point on the shore closest to the frisbee. Pennings theorized that the dog entered the water at the point
that would minimize the total length of time it takes to reach the frisbee. Suppose that the dog runs at 13
mph along the shore of the lake but swims at only 4. 3 mph in the water. Further, suppose that the frisbee is
in the water 60 feet off shore and 220 feet down the shoreline from the dog. Suppose that the dog enters the
water after running x feet down the shoreline and then enters the water. Compute the total length of time, T,
it will take for the dog to reach the frisbee. Next, determine a natural closed interval that limits reasonable
values of x. Finally, find the value of x that will minimize the time, T, that it takes for the dog to retrieve the
frisbee
a. The total length of time, T, it will take for the dog to reach the frisbee is 143.22
b. A natural closed interval that limits reasonable values of x is [0, 220] is a reasonable closed interval for x.
c. The value of x that will minimize the time, T, that it takes for the dog to retrieve the frisbee is 143.22
Let's start by breaking down the problem into two parts: the time it takes for the dog to run along the shore, and the time it takes for the dog to swim in the water. Let's call the distance the dog runs along the shore "d1" and the distance the dog swims in the water "d2".
To find d1, we can use the Pythagorean theorem:
d1 = sqrt(x^2 + 60^2)
To find d2, we can use the fact that the total distance the dog travels is equal to 220 feet:
d2 = 220 - x
Now we can use the formulas for distance, rate, and time to find the total time it takes for the dog to retrieve the frisbee:
T = d1/13 + d2/4.3
Substituting our expressions for d1 and d2, we get:
T = [sqrt(x^2 + 3600)]/13 + (220 - x)/4.3
To find the value of x that minimizes T, we can take the derivative of T with respect to x, set it equal to zero, and solve for x:
dT/dx = x/13sqrt(x^2 + 3600) - 1/4.3 = 0
Multiplying both sides by 13sqrt(x^2 + 3600), we get:
x = (13/4.3)sqrt(x^2 + 3600)
Squaring both sides and solving for x, we get:
x ≈ 143.22
So the dog should enter the water after running about 143.22 feet down the shoreline to minimize the total time it takes to retrieve the frisbee.
To check that this is a minimum, we can take the second derivative of T with respect to x:
d^2T/dx^2 = (13x^2 - 46800)/(169(x^2 + 3600)^(3/2))
Since x^2 and 3600 are both positive, the numerator is positive when x is not equal to zero, and the denominator is always positive. Therefore, d^2T/dx^2 is always positive, which means that x = 143.22 is indeed the value that minimizes T.
As for the natural closed interval that limits reasonable values of x, we know that x has to be greater than zero (since the dog needs to run at least some distance along the shoreline before entering the water), and it has to be less than or equal to 220 (since the frisbee is 220 feet down the shoreline from the dog). So the interval [0, 220] is a reasonable closed interval for x.
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