Can you work out at what height the ladders cross?
Are you surprised that we don't need to know the distance between the
two walls?
Suppose now the first ladder is 10m above the
base of the second wall, with the second ladder
remaining as before.
Can you work out at what height the ladders
cross now?
5m
Suppose the first ladder meets the wall at a
height of a metres, as in the diagram.
At what height do the ladders cross now?
Suppose the second ladder meets the wall at a
height of b metres, as in the diagram.
At what height do the ladders cross now?
5m
nrich.maths.org/510
© University of Cambridge
10m
a

The Vales of x in the equations are : 1) 3 (2) 1  (3) 5  (4) 14  (5) 10 respectively

A linear equation in one variable is that in which the highest power of its variable is 1

The given equations and their solutions are

2x +1 = 7

By collecting like terms we have

2x = 3

x = 3

2) The given equation is 4x +7 = 11

By collecting like terms we have

4x = 4

Making x the subject we have

x = 1

3) 2x -3 = 7

By collecting like terms we have

2x = 10

Making x the subject we have

x = 5

4) 1/2 x - 3 = 7

By collecting like terms we have

1/2x = 7

x= 14

5) The given equation is 3x - 11 = 19

By collecting like terms we have

3x= 30

Therefore x= 10

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The relationship between the cup size and the cost is proportional and the constant of proportionality is 0.3.

At a restaurant, hot chocolate can be purchased in two different cup sizes. A 12-ounce cup costs $3.60 and a 16-ounce cup costs $4.80.

Let's find if the relationships of the variables are proportional.

proportional relationship are relationships between two variables where their ratios are equivalent.

Therefore, proportional relationship is rep[resented as follows:

y = kx

where

Hence,

3.6 = 12k

k = 3.6 / 12

k = 0.3

4.80 = 16k

k = 4.8 / 16

k = 0.3

Therefore, the constant of proportionality is 0.3

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Answer:

The relationship between the cup size and the cost is proportional and the constant of proportionality is 0.3.

Step-by-step explanation:

The computation of the profit and percentage return on Frank Stewart's stock transaction is as follows:

Profits represent the gain generated from an investment or business activity.

The percentage returns are the profits expressed as a percentage of the investment.

The percentage returns are also known as the return on investment (ROI).

Total investment in the common stock of Red Hot Food Processing = $3,100

The sales proceeds after commission = $3,200

Profit = $100 ($3,200 - $3,100)

Percentage return = 3.23% ($100/$3,100 x 100)

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Answer:

TV = 20

Step-by-step explanation:

TV is given by the formula

TV =TU + UV

From the question

TU = 10

AND

UV = 10

Therefore

TV= 10 + 10

TV = 20.

The probability of each combination occurring is 1/15.

Let G be the set of grandchildren, G={Connor, Declan, Lucas, Piper, Sedona, Zayne}.

We can use the formula of nP2 to calculate the possible outcomes of the two grandkids being selected, where n is the number of elements in the set G or 6:

6P2 = 6!/(4!2!) = (6 x 5)/2 = 15

Therefore, there are 15 possible combinations in which the two tickets to the holiday show could go to. These 15 combinations are:

Combination 1: Connor and Declan

Combination 2: Connor and Lucas

Combination 3: Connor and Piper

Combination 4: Connor and Sedona

Combination 5: Connor and Zayne

Combination 6: Declan and Lucas

Combination 7: Declan and Piper

Combination 8: Declan and Sedona

Combination 9: Declan and Zayne

Combination 10: Lucas and Piper

Combination 11: Lucas and Sedona

Combination 12: Lucas and Zayne

Combination 13: Piper and Sedona

Combination 14: Piper and Zayne

Combination 15: Sedona and Zayne

The probability of each combination occurring is 1/15.

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variables is a nominal, ordinal, discrete, or continuous variable.Nominal: Labels/categories. Ordinal: Order/ranking. Discrete: Whole numbers. Continuous: Any value in range

Nominal variable: A nominal variable is a type of variable that is used to name, label, or categorize particular attributes that are being measured. It is often used in survey research to represent different categories or groups.

Ordinal variable: An ordinal variable is a type of variable that is used to represent a specific order or ranking of values. It is frequently used when measuring psychological or social characteristics, such as job satisfaction or political opinions.

Discrete variable: A discrete variable is a type of variable that can only take on certain values, such as whole numbers. It is often used to count the number of specific events or occurrences within a given time period.

Continuous variable: A continuous variable is a type of variable that can take on any value within a given range. It is typically used to measure quantity or amount, such as the height or weight of an individual.

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The meaning of the interval is "there has been no change in support for the candidate". Thus, option (a) is correct.

The 90% two-proportion z-confidence interval (-0.0199, 0.0510) represents the likely range of the genuine difference between the two polls in support of candidate X.

Because the interval comprises 0, the difference in support between the two polls is unlikely to be statistically significant. In other words, there is no compelling evidence that there has been a significant shift in support for the candidate.

Options b), c), d), and e) imply a substantial change in support for the candidate, either positive or negative.

Thus, option (a) is correct.

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Answer:

D

Step-by-step explanation:

The statement says that Evan sells shirts for $10 and after selling 10 shirts, he has $70. We know that the profit is the revenue (the amount of money earned) minus the costs. In this case, the revenue is the amount of money Evan earns from selling shirts and the costs are the amount of money Evan has spent on producing and buying the shirts. We can use this information to set up the equation y = 10x - c where y is the profit, x is the number of shirts sold, and c is the costs.

Given that Evan has $70 after selling 10 shirts, we can substitute 10 for x and 70 for y into the equation to get:

70 = 10(10) - c

Solving for c we get:

c = -30

This means that the costs were $30. To find out how much money Evan had before he sold any shirts we need to subtract the costs from the initial amount. Evan had $0 before he sold any shirts, so the answer is D) $0.

The equation that relates x and y is: x = 3y/ 5

A direct proportion is a variation in which two given variables are related to one another in such a way that as the value of one increases, the values of the other also increases.

Thus, given that x and y are proportional, we have;

x [tex]\alpha[/tex] y

x = ky

But x = 12, and y = 20. Then;

12 = k*20

k = 12/ 20

  = 3/5

So that,

substituting the value of k, to have

x = 3/5 * y

  = 3y/ 5

x = 3y/ 5

Therefore, the equation that relates x and y is: x = 3y/ 5

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Answer:

405.37

Step-by-step explanation:

tan(67) = 955/ X

(tan(67) )(X), (955) (X)

X tan(67) = 955

X tan(67)/ tan(67) = 955/ tan(67)

X= 405.37

The result of the given expression in the scientific notation is 2.7 × 10⁻⁸.

What is the scientific notation answer?

Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10⁸.

To compute the expression (3.0 × 10⁻³)³, we can use the following steps.

Multiply 3.0 × 10⁻³ by itself three times.

The expression (3.0 × 10⁻³)³ can be calculated as follows:

(3.0 × 10⁻³)³ = (3.0)³ × (10⁻³)³ = 27 × 10⁻⁹

So the answer in scientific notation is 2.7 × 10⁻⁸.

Therefore, the answer in scientific notation is 2.7 × 10⁻⁸.

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The optimal solution is to assign team 1 to game A, team 2 to game C, and team 3 to game B. The total distance traveled by the officials is 200km.

A) Objective Function:

Minimize Z = 90x1A + 10x1B + 100x1C + 50x2A + 40x2B + 40x2C + 60x3A + 170x3B + 90x3C

Constraints:

x1A + x1B + x1C = 1

x2A + x2B + x2C = 1

x3A + x3B + x3C = 1

x1A, x1B, x1C, x2A, x2B, x2C, x3A, x3B, x3C ≥ 0

B) The optimal solution is as follows:

x1A = 1

x1B = 0

x1C = 0

x2A = 0

x2B = 0

x2C = 1

x3A = 0

x3B = 1

x3C = 0

Z = 200 km

1. Create the objective function

Z = 90x1A + 10x1B + 100x1C + 50x2A + 40x2B + 40x2C + 60x3A + 170x3B + 90x3C

2. Set the constraints

x1A + x1B + x1C = 1

x2A + x2B + x2C = 1

x3A + x3B + x3C = 1

x1A, x1B, x1C, x2A, x2B, x2C, x3A, x3B, x3C ≥ 0

3. Solve the linear programming problem

Using QM, the optimal solution is:

x1A = 1

x1B = 0

x1C = 0

x2A = 0

x2B = 0

x2C = 1

x3A = 0

x3B = 1

x3C = 0

4. Calculate the total distance

Z = 90x1A + 10x1B

The optimal solution is to assign team 1 to game A, team 2 to game C, and team 3 to game B. The total distance traveled by the officials is 200km.

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By evaluating ∭ (x + y + z) dV, the final answer is 342.

What is the change of variables method?

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better-understood problem.

We can evaluate the triple integral by using the change of variables method. We can set u = x, v = y, and w = z, and find the corresponding bounds for each variable in terms of the original bounds.

Then, we have

∭ (x + y + z) dV = ∭ (u + v + w) det(J) du dv dw,

where J is the Jacobian matrix of the transformation and det(J) is its determinant.

We know that 0 <= x <= 3, 0 <= y <= 3, and 0 <= z <= 9 - x^2 - y^2. Therefore, we have:

u = x, 0 <= u <= 3

v = y, 0 <= v <= 3

w = z, 0 <= w <= 9 - u^2 - v^2

The determinant of the Jacobian matrix is 1, so we can simplify the integral to:

∭ (u + v + w) du dv dw = ∫(0,3) ∫(0,3) ∫(0,9-u^2-v^2) (u+v+w) dw dv du

= ∫(0,3) ∫(0,3) (u+v)(w) + (1/3)(w^3) dv du

Hence, By solving this iteratively we get the final answer as 342.

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Yes there is difference in the national origins of cars driven by students and staff, and that is 31 cars.

Following is the table showing the data of the survey of auto parked in the student lot and the staff lot, classified the brands by the country of origin. we can see that total number of car of all the given countries, driven by the students is 195. And the total number of car of all the given countries, driven by the staff is 164.

So the difference is 195 - 164 = 31 cars.

--The given question is incomplete, the complete question is

"A random survey of autos parked in the student lot and the staff lot at a large university classified the brands by country of origin, as seen in the accompanying table. Are there any difference in the national origins of cars driven by students and staff?

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The equation for an exponential function is f(x) = [tex](ab)^{x}[/tex], where a and b are constants, and the value of f(12) is 1144.14.

An exponential function is one that raises the independent variable to a power. It is typically stated as f(x) = [tex]b^{x}[/tex], where b is the base and x is the exponent. Several real-world phenomena, such as population expansion, radioactive decay, and compound interest, are modelled using exponential functions.

The equation for an exponential function is f(x) = [tex](ab)^{x}[/tex], where a and b are constants.

Since we know that f(3.5) = 10 and f(9.5) = 30, we can solve for a and b.

We first need to solve for b. We can use the equation: b = f(x) / f(x-1)

So, b = 30 / 10 = 3

Now, we can solve for a. We can use the equation: a = f(x) / [tex]b^{x}[/tex]

So, a = 10 / [tex](3)^{3.5}[/tex] = 0.093

Now, we can plug in the values for a and b and x = 12 into our equation to find f(12).

f(12) = 0.093 * [tex]3^{12}[/tex] = 1144.14

Therefore, the value of f(12), to the nearest hundredth, is 1144.14.

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The amount of average cost 50 years ago will be $2,667.05.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.

The percentage is given as,

Percentage (P) = [Final value - Initial value] / Initial value x 100

Over the last 50 years, the average cost of a car has increased by a total of 1,126%. If the average cost of a car today is 30,031. Then the amount of  average cost 50 years ago will be given as,

⇒ $30,031 / 1,126%

⇒ $30,031 / 11.26

⇒ $2,667.05

The amount of average cost 50 years ago will be $2,667.05.

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Answer:

1 inch = 9 yards

Step-by-step explanation:

a must be equal to zero for the given equation to be true.

The given polynomial can be written in the form [tex]$(x-1)(x+1) = 0$[/tex]. This expression can be equated to zero to solve for $x$. By equating it to zero and solving, we can find that [tex]$x = 1$[/tex] and [tex]$x = -1$[/tex]. We can substitute each of these values into the given equation and solve for $a$.

Forx = 1:

[tex]$a(1)^{17}b(1)^{16}1 = 0$ \\ $a\cdot 1 \cdot b \cdot 1 \cdot 1 = 0$\\$a \cdot b = 0$[/tex]

Therefore, $a$ must be equal to zero for the given equation to be true.

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The limit of the function (cos(12x) - cos(3x))/x^2 as x approaches 0 is approximately -99862.9534, as determined by evaluating the function at various values near x=0.

Let f(x) : (cos(12x) - cos(3x))/x^2

We want to find the limit lim x=0 from (cos(12x) - cos(3x))/x^2

Step 1: Calculate the value of the function for x=0.2

fx = (cos(12(0.2)) - cos(3(0.2)))/(0.2)^2

fx = (cos(2.4) - cos(0.6))/0.04

fx = (0.92358 - 0.95105)/0.04

fx = -24.987664

Step 2: Calculate the value of the function for x=0.1

fx = (cos(12(0.1)) - cos(3(0.1)))/(0.1)^2

fx = (cos(1.2) - cos(0.3))/0.01

fx = (0.9589 - 0.9511)/0.01

fx = -98.998848

Step 3: Calculate the value of the function for x=0.05

fx = (cos(12(0.05)) - cos(3(0.05)))/(0.05)^2

fx = (cos(0.6) - cos(0.15))/0.0025

fx = (0.9802 - 0.9657)/0.0025

fx = -19.923683

Step 4: Calculate the value of the function for x=0.01

fx = (cos(12(0.01)) - cos(3(0.01)))/(0.01)^2

fx = (cos(0.12) - cos(0.03))/0.0001

fx = (0.9999 - 0.9998)/0.0001

fx = -99.853172

Step 5: Calculate the value of the function for x=0.001

fx = (cos(12(0.001)) - cos(3(0.001)))/(0.001)^2

fx = (cos(0.012) - cos(0.003))/0.000001

fx = (0.999983 - 0.999971)/0.000001

fx = -998.62855

Step 6: Calculate the value of the function for x=0.0001

fx = (cos(12(0.0001)) - cos(3(0.0001)))/(0.0001)^2

fx = (cos(0.0012) - cos(0.0003))/0.00000001

fx = (0.99999983 - 0.99999971)/0.00000001

fx = -9989.29525

Step 7: Calculate the value of the function for x=0.00001

fx = (cos(12(0.00001)) - cos(3(0.00001)))/(0.00001)^2

fx = (cos(0.00012) - cos(0.00003))/0.0000000001

fx = (0.9999999983 - 0.9999999971)/0.0000000001

fx = -99862.9534

Therefore, the limit of the function

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Answer:

288

Solution

(8x5)x4=160

(8x8)x2=64

160+64=288

Answer: the answer I don’t know

Step-by-step explanation:

Answer:

-2.29

Step-by-step explanation:

Slope is Y - Y/X - X

- 5 - 11/ - 1 - (- 8)

-16/-1+8

-16/7

-2.29

Two methods can be used to administer the test.

Therefore, We infer that the solution to these questions is option C. The likelihood of noticing a difference of 0.006 is 0.018 if there is no difference in the proportions of all defective parts produced on the two assembly lines.

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Answer:

The answer would be [tex]x[/tex]=17 because the food cost 85% of Andy's bill, that is, $17.

Step-by-step explanation:

Since 15% of the $20 bill Andy paid was for drink, it means that (100−15)% of the same amount will be for food. Hence, the food came to 85% of the bill.The amount (x) Andy paid for food therefore is:

[tex]x[/tex]= 20 x 85/100

[tex]x[/tex]= 1 x 85/5

[tex]x[/tex] = 85/5

[tex]x[/tex] = 17

So, the answer would be 17$.

Answer:

M

Step-by-step explanation:

2,462 rounded is 2,500

2.5 × 10^3= 2,500

The answer is 2.5 × 10³ Because its = 2500

Answer:

slope is 4

Step-by-step explanation:

A wire that supports telegraph poles is joined to them at intervals of 6 metres above the ground and 2.5 metres from the pole's foot for a total length of 7.3 metres.

Assume that x is the wire's length. [tex]$x^2=2.5^2+6^2$[/tex] Pythagorean theory [tex]$x^2=6.25+36$[/tex]

[tex]$x^2=6.25+36$\\ $x^2=42.25$[/tex]

As [tex]$x > 0, x=\sqrt{42,2 \sqrt{2}}=6, \sqrt{m}$[/tex]

Total length [tex]$=6.5+2.8=7.3 \mathrm{~m}$[/tex]

There are 50 metres between each telephone pole. Utility poles are made more stable by guy wires. The pole is covered by a ground wire the entire way around. Any power on the pole is safely directed underground by this device. This is a representation of the essential hardware that can differ depending on where you are found on a distribution pole. Typically, lineside telegraph poles are placed 60 to 70 yards apart, a little closer, and ideally within curves.

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The lowest possible value of R² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R² will be to 1.

The coefficient of determination (R²) measures how well a statistical model predicts an outcome. The outcome is represented by the model’s dependent variable.

The lowest possible value of R² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R² will be to 1.

Suppose that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying (independent variable).

If the R2 is 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an average exam score.

If the R2 is between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score.

If the R2 is 1, the model allows you to perfectly predict anyone’s exam score.

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The greatest positive acceleration occurs at the midpoint of the interval, when the particle is changing velocity most rapidly.

What is velocity?

Velocity is a measure of the rate of change of an object's position over a period of time. It is a vector quantity, which means it has both magnitude (or size) and direction. Velocity is usually expressed in terms of distance traveled per unit of time, such as meters per second (m/s). Velocity can be determined by dividing the distance traveled by the time taken. For example, if an object travels 20 meters in 5 seconds, its velocity would be 4 m/s. Velocity can also be calculated by taking the derivative of the object's position with respect to time.

The particle will achieve its greatest positive acceleration at the midpoint of the interval, at t = b/2.

This is because the acceleration is greatest when the velocity is changing most rapidly.

At the beginning of the interval, the particle has zero velocity, so it has no acceleration.

At the end of the interval, the particle has reached its maximum velocity, and the rate of change of velocity is zero, so it has no acceleration. Therefore, the greatest positive acceleration occurs at the midpoint of the interval, when the particle is changing velocity most rapidly.

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Answer:

1. x/0=7

2.x+1=2

3. x=7

Step-by-step explanation:

1. you can't divide by zero so it's undefined

2. x=1

3. when you graph this on a graph, the line goes on forever, as long as it satisfies the equation x+1.