Chapter 4

Slippery Koads If a car is moving at 50 miles per hour on a level highway, then its braking distance depends on the road conditions. This distance in feet can be computed by \(D(x)=20 x^{00},\) where \(x\) is the coefficient of friction between the tires and the road and \(0

Assume that the constant of proportionality is positive. Suppose \(y\) is directly proportional to the second power of \(x .\) If \(x\) is halved, what happens to \(y ?\)

If an even function \(f\) has an absolute minimum of \(-6\) at \(x=-2,\) then what else can be said about \(f ?\) Explain.

The concentration of a drug in a medical patient's bloodstream is given by the formula \(f(t)=\frac{5}{t^{2}+1},\) where the input \(t\) is in hours, \(t \geq 0,\) and the output is in milligrams per liter. (a) Does the concentration of the drug increase or decrease? Explain. (b) The patient should not take a second dose until the concentration is below 1.5 milligrams per liter. How long should the patient wait before taking a second dose?

Assume that the constant of proportionality is positive. Suppose \(y\) is directly proportional to the second power of \(x .\) If \(x\) is halved, what happens to \(y ?\) $$ \begin{array}{ccccc} x & 2 & 3 & 4 & 5 \\ \hline y & 2 & 4.5 & 8 & 12.5 \end{array} $$

Let \(f(x)\) be the formula for a rational function. (a) Explain how to find any vertical or horizontal asymptotes of the graph of \(f\) (b) Discuss what a horizontal asymptote represents.

Assume that the constant of proportionality is positive. Suppose \(y\) is directly proportional to the second power of \(x .\) If \(x\) is halved, what happens to \(y ?\) $$ \begin{array}{ccccc} x & 3 & 5 & 7 & 9 \\ \hline y & 32.4 & 150 & 411.6 & 874.8 \end{array} $$

Discuss how to find the domain of a rational function symbolically and graphically.

The data in the table satisfy the equation \(y=\frac{k}{x^{n}},\) where \(n\) is a positive integer. Determine \(k\) and \(n\) $$ \begin{array}{ccccc} x & 2 & 3 & 4 & 5 \\ \hline y & 1.5 & 1 & 0.75 & 0.6 \end{array} $$

The data in the table satisfy the equation \(y=\frac{k}{x^{n}},\) where \(n\) is a positive integer. Determine \(k\) and \(n\) $$ \begin{array}{ccccc} x & 2 & 4 & 6 & 8 \\ \hline y & 9 & 2.25 & 1 & 0.5625 \end{array} $$

The weight \(y\) of a fiddler crab is directly proportional to the 1.25 power of the weight \(x\) of its claws. A crab with a body weight of 1.9 grams has claws weighing 1.1 grams. Estimate the weight of a fiddler crab with claws weighing 0.75 gram. (Source: D. Brown.)

The weight of an object varies inversely as the second power of the distance from the center of Earth. The radius of Earth is approximately 4000 miles. If a person weighs 160 pounds on Earth's surface, what would this individual weigh 8000 miles above the surface of Earth?

The brightness, or intensity, of starlight varies inversely as the square of its distance from Earth. The Hubble Telescope can see stars whose intensities are \(\frac{1}{50}\) that of the faintest star now seen by ground-based telescopes. Determine how much farther the Hubble Telescope can see into space than ground based telescopes. (Sounce: National Aeronautics and Space Administration.)

The volume \(V\) of a cylinder with a fixed height is directly proportional to the square of its radius \(r\). If a cylinder with a radius of 10 inches has a volume of 200 cubic inches, what is the volume of a cylinder with the same height and a radius of 5 inches?

The electrical resistance \(R\) of a wire varies inversely as the square of its diameter \(d\). If a 25-foot wire with a diameter of 2 millimeters has a resistance of 0.5 ohm, find the resistance of a wire having the same length and a diameter of 3 millimeters.

The strength of a rectangular wood beam varies directly as the square of the depth of its cross section. If a beam with a depth of 3.5 inches can support 1000 pounds, how much weight can the same type of beam hold if its depth is 12 inches?

The frequency \(F\) of \(a\) vibrating string is directly proportional to the square root of the tension \(T\) on the string and inversely proportional to the length \(L\) of the string. If both the tension and the length are doubled, what happens to \(F ?\)

The frequency \(F\) of \(a\) vibrating string is directly proportional to the square root of the tension \(T\) on the string and inversely proportional to the length \(L\) of the string. Give two ways to double the frequency \(F\).