Chapter 7

$$ \text { Solve the given quadratic equations by factoring.}$$ In determining the speed \(s\) (in \(\mathrm{mi} / \mathrm{h}\) ) of a car while studying its fuel economy, the equation \(s^{2}-16 s=3072\) is used. Find \(s\)

Solve the given applied problem. The diagonal of a rectangular floor is \(3.00 \mathrm{ft}\) less than twice the length of one of the sides. If the other side is \(15.0 \mathrm{ft}\) long, what is the area of the floor?

Solve the given problems. All numbers are accurate to at least two significant digits. Solve \(6 x^{2}-x=15\) for \(x\) by (a) factoring, (b) completing the square, and (c) the quadratic formula. Which is (a) longest?, (b) shortest?

Solve the given quadratic equations by factoring.In determining the speed \(s\) (in \(\mathrm{mi} / \mathrm{h}\) ) of a car while studying its fuel economy, the equation \(s^{2}-16 s=3072\) is used. Find \(s\).

Find the indicated quadratic equations. Find a quadratic equation for which the solutions are 0.5 and 2

Solve the given problems. All numbers are accurate to at least two significant digits. In machine design, in finding the outside diameter \(D_{0}\) of a hollow shaft, the equation \(D_{0}^{2}-D D_{0}-0.25 D^{2}=0\) is used. Solve for \(D_{0}\) if \(\bar{D}=3.625 \mathrm{cm}\).

Solve the given applied problem. A security fence is to be built around a rectangular parking area of \(20,000 \mathrm{ft}^{2}\). If the front side of the fence costs \(\$ 20 / \mathrm{ft}\) and the other three sides cost \(\$ 10 / \mathrm{ft}\), solve graphically for the dimensions (to \(1 \mathrm{ft}\) ) of the parking area if the fence is to cost \(\$ 7500 .\) See Fig. 7.18. Figure cannot copy

In Exercises 47 and \(48,\) find the indicated quadratic equations.Find a quadratic equation for which the solutions are 0.5 and 2 .

Solve the given applied problem. An airplane pilot could decrease the time \(t\) (in h) needed to travel the 630 mi from Ottawa to Milwaukee by 20 min if the plane's speed \(v\) is increased by \(40 \mathrm{mi} / \mathrm{h}\). Set up the appropriate equation and solve graphically for \(v\) (to two significant digits).

Solve the given problems. All numbers are accurate to at least two significant digits. A missile is fired vertically into the air. The distance \(s\) (in \(\mathrm{ft}\) ) above the ground as a function of time \(t\) (in s) is given by \(s=300+500 t-16 t^{2} .\) (a) When will the missile hit the ground? (b) When will the missile be 1000 ft above the ground?

Although the equations are not quadratic, factoring will lead to one quadratic factor and the solution can be completed by factoring as with a quadratic equation. Find the three roots of each equation. $$x^{3}-x=0$$

Solve the given problems. All numbers are accurate to at least two significant digits. In analyzing the deflection of a certain beam, the equation \(8 x^{2}-15 L x+6 L^{2}\) is used. Solve for \(x,\) if \(x

Although the equations are not quadratic, factoring will lead to one quadratic factor and the solution can be completed by factoring as with a quadratic equation. Find the three roots of each equation. $$x^{3}-4 x^{2}-x+4=0$$

Solve the given problems. All numbers are accurate to at least two significant digits. A homeowner wants to build a patio with an area of \(20.0 \mathrm{m}^{2},\) such that the length is \(2.0 \mathrm{m}\) more than the width. What should the dimensions be?

Solve the given equations involving fractions. $$\frac{1}{x-3}+\frac{4}{x}=2$$

Solve the given problems. All numbers are accurate to at least two significant digits. Two cars leave an intersection at the same time, one going due east and the other due south. After one has gone \(2.0 \mathrm{km}\) farther than the other, they are \(6.0 \mathrm{km}\) apart on a direct line. How far did each go?

Solve the given equations involving fractions. $$2-\frac{1}{x}=\frac{3}{x+2}$$

Solve the given problems. All numbers are accurate to at least two significant digits. A student cycled \(3.0 \mathrm{km} / \mathrm{h}\) faster to college than when returning, which took 15 min longer. If the college is \(4.0 \mathrm{km}\) from home, what were the speeds to and from college?

Solve the given equations involving fractions. $$\frac{1}{2 x}-\frac{3}{4}=\frac{1}{2 x+3}$$

Solve the given problems. All numbers are accurate to at least two significant digits. For a rectangle, if the ratio of the length to the width equals the ratio of the length plus the width to the length, the ratio is called the golden ratio. Find the value of the golden ratio, which the ancient Greeks thought had the most pleasing properties to look at.

Solve the given equations involving fractions. $$\frac{x}{2}+\frac{1}{x-3}=3$$

Solve the given problems. All numbers are accurate to at least two significant digits. When focusing a camera, the distance \(r\) the lens must move from the infinity setting is given by \(r=f^{2} /(p-f),\) where \(p\) is the distance from the object to the lens, and \(f\) is the focal length of the lens. Solve for \(f\).

Solve the given problems. All numbers are accurate to at least two significant digits. In calculating the current in an electric circuit with an inductance \(L,\) a resistance \(R,\) and a capacitance \(C,\) it is necessary to solve the equation \(L m^{2}+R m+1 / C=0 .\) Solve for \(m\) in the terms of \(L\) \(R,\) and \(C .\) See Fig. 7.7

Set up the appropriate quadratic equations and solve.The spring constant \(k\) is the force \(F\) divided by the amount \(x\) the spring stretches \((k=F / x) .\) See Fig. \(7.2(\) a). For two springs in series (see Fig. \(7.2(b)\) ), the reciprocal of the spring constant \(k_{c}\) for the combination equals the sum of the reciprocals of the individual spring constants. Find the spring constants for each of two springs in series if \(k_{c}=2 \mathrm{N} / \mathrm{cm}\) and one spring constant is \(3 \mathrm{N} / \mathrm{cm}\) more than the other.

Solve the given problems. All numbers are accurate to at least two significant digits. In finding the radius \(r\) of a circular arch of height \(h\) and span \(b,\) we use the formula shown at the right. Solve for \(h\). $$r=\frac{b^{2}+4 h^{2}}{8 h}$$

Set up the appropriate quadratic equations and solve.The reciprocal of the combined resistance \(R\) of two resistances \(R_{1}\) and \(R_{2}\) connected in parallel (see Fig. \(7.3(a)\) ) is equal to the sum of the reciprocals of the individual resistances. If the two resistances are connected in series (see Fig. \(7.3(\mathrm{b})\) ), their combined resistance is the sum of their individual resistances. If two resistances connected in parallel have a combined resistance of \(3.0 \Omega\) and the same two resistances have a combined resistance of \(16 \Omega\) when connected in series, what are the resistances?

Set up the appropriate quadratic equations and solve. A hydrofoil made the round-trip of \(120 \mathrm{km}\) between two islands in 3.5 h of travel time. If the average speed going was \(10 \mathrm{km} / \mathrm{h}\) less than the average speed returning, find these speeds.

Solve the given problems. All numbers are accurate to at least two significant digits. A computer monitor has a viewing screen that is \(33.8 \mathrm{cm}\) wide and \(27.3 \mathrm{cm}\) high, with a uniform edge around it. If the edge covers \(20.0 \%\) of the monitor front, what is the width of the edge?

Set up the appropriate quadratic equations and solve. A rectangular solar panel is \(20 \mathrm{cm}\) by \(30 \mathrm{cm} .\) By adding the same amount to each dimension, the area is doubled. How much is added?

Solve the given problems. All numbers are accurate to at least two significant digits. An investment of 2000 dollar is deposited at a certain annual interest rate. One year later, 3000 dollar is deposited in another account at the same rate. At the end of the second year, the accounts have a total value of 5319.05 dollar . What is the interest rate?

Solve the given problems. All numbers are accurate to at least two significant digits. In remodeling a house, an architect finds that by adding the same amount to each dimension of a 12 -ft by 16 -ft rectangular room, the area would be increased by \(80 \mathrm{ft}^{2}\). How much must be added to each dimension?

Solve the given problems. All numbers are accurate to at least two significant digits. Two pipes together drain a wastewater-holding tank in \(6.00 \mathrm{h}\). If used alone to empty the tank, one takes \(2.00 \mathrm{h}\) longer than the other. How long does each take to empty the tank if used alone?

Solve the given problems. All numbers are accurate to at least two significant digits. On some California highways, a car can legally travel \(15 \mathrm{mi} / \mathrm{h}\) faster than a truck. Traveling at maximum legal speeds, a car can travel \(77 \mathrm{mi}\) in 18 min less than a truck. What are the maximum legal speeds for cars and for trucks?

Solve the given problems. All numbers are accurate to at least two significant digits. For an optical lens, the sum of the reciprocals of \(p,\) the distance of the object from the lens, and \(q\), the distance of the image from the lens, equals the reciprocal of \(f,\) the focal length of the lens. If \(p\) is \(5.0 \mathrm{cm}\) greater than \(q(q>0)\) and \(f=-4.0 \mathrm{cm},\) find \(p\) and \(q\).

Solve the given problems. All numbers are accurate to at least two significant digits. The length of a tennis court is \(12.8 \mathrm{m}\) more than its width. If the area of the tennis court is \(262 \mathrm{m}^{2},\) what are its dimensions? See Fig. 7.8