Two protons will repel each other with a force F that depends on the distance d between them. Some values of F are given in the table given below. Note that d is measured in $10^{-10}$ meters, and F is measured in $10^{-10}$ newtons.

a. Graph the given data.

b. Find a general variation equation to represent the situation.

c. Find the value of the constant of variation and use it to rewrite the variation equation.

d. Use your variation equation from Part c to answer the question.

With what force will two protons repel each other if they are $1 \times 10^{-10}{~m}$ apart?

$$\begin{array}{c|ccccc} d & 1.5 & 3 & 4.5 & 6 & 7.5 \\ \hline F & 103 & 26 & 11 & 6 & 4 \end{array}$$

Fill in the Blank space in the following problem.

If y = kx$^n$, and x is divided by c, then y is _______.

Fill in the Blanks in the following sentence.

If $y=\frac{k x}{v^2}$, where k is a constant, then y varies _____ as _____, and varies ______ as _____.

For the variation equation given below, how does the y-value change when an x-value is doubled? Give two specific ordered pairs (x, y) that support your conclusion.

$$y=\frac{5}{x^2}$$

For this question, consider the function f with equation $f(x)=\frac{15}{x}$.

a. Evaluate f(-10) and f(15).

b. Graph f in the window $-12 \leq x \leq 12$ and $-12 \leq y \leq 12$ and sketch the graph.

Sonia calculated the volume of a cylindrical can of cat food with a radius of 1.5 inches and a height of 1.2 inches to be about 9.2 cubic inches.

a. Use the equation $V=k r^2h$ to calculate the constant of variation k that she used.

b. Did Sonia use the correct formula? How do you know?

Officer Friendly measured the length L of car skid marks when the brakes were applied at different starting speeds S. He obtained the following data.

$$\begin{array}{l|ccccc} S(\mathrm{kph}) & 40 & 60 & 80 & 100 & 120 \\ \hline L(\mathrm{~m}) & 9 & 20 & 36 & 56 & 81 \end{array}$$

a. Graph the given data.

b. Find a general variation equation to represent the situation.

c. Find the value of the constant of variation and use it to rewrite the variation equation.

d. Use your variation equation from Part c to answer the question.

How far would a car skid if the brakes were applied at 150 kph ?

An object is tied to a string and then twirled in a circular motion. The tension in the string varies directly as the square of the speed of the object and inversely as the length of the string. When the length of the string is 2 ft and the speed is 3 ft/sec, the tension on the string is 130 lb. If the string is shortened to 1.5 ft and the speed is increased to 3.4 ft/sec, find the tension on the string.

The force required to prevent a car from skidding on a flat curve varies directly as the weight of the car and the square of its speed, and inversely as the radius of the curve. It requires 290 lb of force to prevent a 2,200 lb car traveling at 35 mph from skidding on a curve of radius 520 ft. How much force is required to keep a 2,800 lb car traveling at 50 mph from skidding on a curve of radius 415 ft ?

A credit card promotion offers a discount on all purchases made with the card. The discount is directly proportional to the size of the purchase. If you save $1.35 on a$20 purchase, how much will you save on a $173 purchase?

Recall that the weight of an object is inversely proportional to the square of its distance from the center of Earth. If you weigh 115 pounds on the surface of Earth, how much would you weigh in space 50,000 miles from Earth's surface? (The radius of Earth is about 4,000 miles.)

One of Murphy's Laws says that the time t spent debating a budget item is inversely proportional to the number d of dollars involved. According to this law, if a committee spends 45 minutes debating a $5,000 item, how much time will they spend debating a$10,000,000 item?

Complete the following sentence with directly, Inversely, or neither directly nor inversely.

Your height above the ground while on a Ferris wheel varies _____ as the number of minutes you have been riding it.

Complete the following sentence with directly, Inversely, or neither directly nor inversely.

The wall area that can be painted with a gallon of paint varies _____ as the thickness of the applied paint.