For the following problem, Suppose k< 0.

State the domain and range of the function $f: x \rightarrow k x^2$.

The lengths of the femur and tibia within a species of mammal are typically directly proportional. If the femur of one household cat is 116.0 mm long and its tibia is 122.8 mm long, how long is the tibia of a cat whose femur is 111.5mm long?

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?

The speed of sound in air is about 1,088 feet per second.

a. Convert the speed of sound to miles per second.

b. Write the relationship between the number of miles sound travels and the number of seconds it takes sound to travel that distance as a direct variation. Be sure to identify what your variables represent.

c. Use your answer from Part b to find the time it takes sound to travel four miles in air.

When lightning strikes in the distance, you do not see the flash and hear the thunder at the same time. You first see the lightning, then you hear the thunder.

a. Write an equation to represent the following situation: The distance d (in miles) from the observer to the flash varies directly as the time t (in seconds) between seeing the lightning and hearing the thunder.

b. Suppose that lightning strikes a point 4 miles away and that you hear the thunder 20 seconds later. How far away has lightning struck if 12 seconds pass between seeing a flash and hearing its thunder?

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.

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

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 _____.

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.