Chapter 3

Exer. 53-60: Find a composite function form for \(y\). $$ y=\sqrt[4]{x^{4}-16} $$

A travel agency offers group tours at a rate of $$\$ 60$$ per person for the first 30 participants. For larger groups - up to 90 -each person receives a $$\$ 0.50$$ discount for every participant in excess of 30 . For example, if 31 people participate, then the cost per person is $$\$ 59.50$$. Determine the size of the group that will produce the maximum amount of money for the agency.

Newborn blue whales are approximately 24 feet long and weigh 3 tons. Young whales are nursed for 7 months, and by the time of weaning they often are 53 feet long and weigh 23 tons. Let \(L\) and \(W\) denote the length (in feet) and the weight (in tons), respectively, of a whale that is \(t\) months of age. (a) If \(L\) and \(t\) are linearly related, express \(L\) in terms of \(t\). (b) What is the daily increase in the length of a young whale? (Use 1 month \(=30\) days.) (c) If \(W\) and \(t\) are linearly related, express \(W\) in terms of \(t\). (d) What is the daily increase in the weight of a young whale?

Exer. 53-54: If a linear function \(f\) satisfies the given conditions, find \(f(x)\). $$ f(-2)=7 \text { and } f(4)=-2 $$

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}-6 x+4 y+13=0 $$

Exer. 55-56: Explain why the graph of the equation is not the graph of a function. $$ x=y^{2} $$

Exer. 53-60: Find a composite function form for \(y\). $$ y=\frac{1}{(x-3)^{4}} $$

A cable television firm presently serves 8000 households and charges $$\$ 50$$ per month. A marketing survey indicates that each decrease of $$\$ 5$$ in the monthly charge will result in 1000 new customers. Let \(R(x)\) denote the total monthly revenue when the monthly charge is \(x\) dollars. (a) Determine the revenue function \(R\). (b) Sketch the graph of \(R\) and find the value of \(x\) that results in maximum monthly revenue.

Suppose a major league baseball player has hit 5 home runs in the first 14 games, and he keeps up this pace throughout the 162 -game season. (a) Express the number \(y\) of home runs in terms of the number \(x\) of games played. (b) How many home runs will the player hit for the season?

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}-2 x-8 y+19=0 $$

Exer. 55-56: Explain why the graph of the equation is not the graph of a function. $$ x=-|y| $$

Exer. 53-60: Find a composite function form for \(y\). $$ y=4+\sqrt{x^{2}+1} $$

A real estate company owns 218 efficiency apartments, which are fully occupied when the rent is $$\$ 940$$ per month. The company estimates that for each $$\$ 25$$ increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income?

A cheese manufacturer produces 18,000 pounds of cheese from January 1 through March \(24 .\) Suppose that this rate of production continues for the remainder of the year. (a) Express the number \(y\) of pounds of cheese produced in terms of the number \(x\) of the day in a 365 -day year. (b) Predict, to the nearest pound, the number of pounds produced for the year.

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}+4 x+6 y+16=0 $$

Exer. 53-60: Find a composite function form for \(y\). $$ y=\left(x^{4}-2 x^{2}+5\right)^{5} $$

When engineers plan highways, they must design hills so as to ensure proper vision for drivers. Hills are referred to as crest vertical curves. Crest vertical curves change the slope of a highway. Engineers use a parabolic shape for a highway hill, with the vertex located at the top of the crest. Two roadways with different slopes are to be connected with a parabolic crest curve. The highway passes through the points \(A(-800,-48), B(-500,0), C(0,40)\), \(D(500,0)\), and \(E(800,-48)\), as shown in the figure. The roadway is linear between \(A\) and \(B\), parabolic between \(B\) and \(D\), and then linear between \(D\) and \(E\). Find a piecewisedefined function \(f\) that models the roadway between the points \(A\) and \(E\).

A baby weighs 10 pounds at birth, and three years later the child's weight is 30 pounds. Assume that childhood weight \(W\) (in pounds) is linearly related to age \(t\) (in years). (a) Express \(W\) in terms of \(t\). (b) What is \(W\) on the child's sixth birthday? (c) At what age will the child weigh 70 pounds? (d) Sketch, on a \(t W\)-plane, a graph that shows the relationship between \(W\) and \(t\) for \(0 \leq t \leq 12\).

Exer. 57-60: Find equations for the upper half, lower half, right half, and left half of the circle. $$ x^{2}+y^{2}=36 $$

Exer. 53-60: Find a composite function form for \(y\). $$ y=\frac{1}{\left(x^{2}+3 x-5\right)^{3}} $$

A college student receives an interestfree loan of \(\$ 8250\) from a relative. The student will repay \(\$ 125\) per month until the loan is paid off. (a) Express the amount \(P\) (in dollars) remaining to be paid in terms of time \(t\) (in months). (b) After how many months will the student owe \(\$ 5000\) ? (c) Sketch, on a \(t P\)-plane, a graph that shows the relationship between \(P\) and \(t\) for the duration of the loan.

Exer. 57-60: Find equations for the upper half, lower half, right half, and left half of the circle. $$ (x+3)^{2}+y^{2}=64 $$

Exer. 59-62: Sketch the graph of the equation. $$ y=\left|9-x^{2}\right| $$

Exer. 53-60: Find a composite function form for \(y\). $$ y=\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2} $$

The amount of heat \(H\) (in joules) required to convert one gram of water into vapor is linearly related to the temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) of the atmosphere. At \(10^{\circ} \mathrm{C}\) this conversion requires 2480 joules, and each increase in temperature of \(15^{\circ} \mathrm{C}\) lowers the amount of heat needed by 40 joules. Express \(H\) in terms of \(T\).

Exer. 57-60: Find equations for the upper half, lower half, right half, and left half of the circle. $$ (x-2)^{2}+(y+1)^{2}=49 $$

Exer. 59-62: Sketch the graph of the equation. $$ y=\left|x^{3}-1\right| $$

Exer. 53-60: Find a composite function form for \(y\). $$ y=\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}} $$

In exercise physiology, aerobic power \(P\) is defined in terms of maximum oxygen intake. For altitudes up to 1800 meters, aerobic power is optimal-that is, \(100 \%\). Beyond 1800 meters, \(P\) decreases linearly from the maximum of \(100 \%\) to a value near \(40 \%\) at 5000 meters. (a) Express aerobic power \(P\) in terms of altitude \(h\) (in meters) for \(1800 \leq h \leq 5000\). (b) Estimate aerobic power in Mexico City (altitude: 2400 meters), the site of the 1968 Summer Olympic Games.

Exer. 57-60: Find equations for the upper half, lower half, right half, and left half of the circle. $$ (x-3)^{2}+(y-5)^{2}=4 $$

Exer. 59-62: Sketch the graph of the equation. $$ y=|\sqrt{x}-1| $$

If \(f(x)=\sqrt{x}-1\) and \(g(x)=x^{3}+1\), approximate \((f \circ g)(0.0001)\). In order to avoid calculating a zero value for \((f \circ g)(0.0001)\), rewrite the formula for \(f \circ g\) as $$ \frac{x^{3}}{\sqrt{x^{3}+1}+1} \text {. } $$

The urban heat island phenomenon has been observed in Tokyo. The average temperature was \(13.5^{\circ} \mathrm{C}\) in 1915 , and since then has risen \(0.032^{\circ} \mathrm{C}\) per year. (a) Assuming that temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) is linearly related to time \(t\) (in years) and that \(t=0\) corresponds to 1915 , express \(T\) in terms of \(t\). (b) Predict the average temperature in the year \(2010 .\)

Exer. 59-62: Sketch the graph of the equation. $$ y=|| x|-1| $$

If \(f(x)=\frac{x^{3}}{x^{2}+x+2}\) and \(g(x)=\left(\sqrt{3 x}-x^{3}\right)^{3 / 2}\), approximate $$ \frac{(f+g)(1.12)-(f / g)(1.12)}{[(f \circ f)(5.2)]^{2}} $$

In 1870 the average ground temperature in Paris was \(11.8^{\circ} \mathrm{C}\). Since then it has risen at a nearly constant rate, reaching \(13.5^{\circ} \mathrm{C}\) in \(1969 .\) (a) Express the temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) in terms of time \(t\) (in years), where \(t=0\) corresponds to the year 1870 and \(0 \leq t \leq 99\). (b) During what year was the average ground temperature \(12.5^{\circ} \mathrm{C}\) ?

Let \(y=f(x)\) be a function with domain \(D=[-2,6]\) and range \(R=[-4,8]\). Find the domain \(D\) and range \(R\) for each function. Assume \(f(2)=8\) and \(f(6)=-4\). (a) \(y=-2 f(x)\) (b) \(y=f\left(\frac{1}{2} x\right)\) (c) \(y=f(x-3)+1\) (d) \(y=f(x+2)-3\) (e) \(y=f(-x)\) (f) \(y=-f(x)\) (g) \(y=f(|x|)\) (h) \(y=|f(x)|\)

The owner of an ice cream franchise must pay the parent company \(\$ 1000\) per month plus \(5 \%\) of the monthly revenue \(R\). Operating cost of the franchise includes a fixed cost of \(\$ 2600\) per month for items such as utilities and labor. The cost of ice cream and supplies is \(50 \%\) of the revenue. (a) Express the owner's monthly expense \(E\) in terms of \(R\). (b) Express the monthly profit \(P\) in terms of \(R\). (c) Determine the monthly revenue needed to break even.

Let \(y=f(x)\) be a function with domain \(D=[-6,-2]\) and range \(R=[-10,-4]\). Find the domain \(D\) and range \(R\) for each function. (a) \(y=\frac{1}{2} f(x)\) (b) \(y=f(2 x)\) (c) \(y=f(x-2)+5\) (d) \(y=f(x+4)-1\) (e) \(y=f(-x)\) (f) \(y=-f(x)\) (g) \(y=f(|x|)\) (h) \(y=|f(x)|\)

Pharmacological products must specify recommended dosages for adults and children. Two formulas for modification of adult dosage levels for young children are \(\begin{array}{lll} & \text { Cowling's rule: } & y=\frac{1}{24}(t+1) a \\\ \text { and } & \text { Friend's rule: } & y=\frac{2}{25} t a,\end{array}\) where \(a\) denotes adult dose (in milligrams) and \(t\) denotes the age of the child (in years). (a) If \(a=100\), graph the two linear equations on the same coordinate plane for \(0 \leq t \leq 12\). (b) For what age do the two formulas specify the same dosage?

A certain country taxes the first \(\$ 20,000\) of an individual's income at a rate of \(15 \%\), and all income over \(\$ 20,000\) is taxed at \(20 \%\). Find a piecewise-defined function \(T\) that specifies the total tax on an income of \(x\) dollars.

In the video game shown in the figure, an airplane flies from left to right along the path given by \(y=1+(1 / x)\) and shoots bullets in the tangent direction at creatures placed along the \(x\)-axis at \(x=1,2,3,4\). shear is of great importance to pilots during takeoffs and landings. If the wind speed is \(v_{1}\) at height \(h_{1}\) and \(v_{2}\) at height \(h_{2}\), then the average wind shear \(s\) is given by the slope formula $$ s=\frac{v_{2}-v_{1}}{h_{2}-h_{1}} $$ If the wind speed at ground level is \(22 \mathrm{mi} / \mathrm{hr}\) and \(s\) has been determined to be \(0.07\), find the wind speed 185 feet above the ground.

From a rectangular piece of cardboard having dimensions 20 inches \(\times 30\) inches, an open box is to be made by cutting out an identical square of area \(x^{2}\) from each corner and turning up the sides (see the figure). Express the volume \(V\) of the box as a function of \(x\).

Exer. 65-66: Determine whether the point \(P\) is inside, outside, or on the circle with center \(C\) and radius \(r\). (a) \(P(2,3), \quad C(4,6), \quad r=4\) (b) \(P(4,2), \quad C(1,-2), \quad r=5\) (c) \(P(-3,5), \quad C(2,1), \quad r=6\)

A certain state taxes the first \(\$ 500,000\) in property value at a rate of \(1 \%\); all value over \(\$ 500,000\) is taxed at \(1.25 \%\). Find a piecewise- defined function \(T\) that specifies the total tax on a property valued at \(x\) dollars.

The relationship between the temperature reading \(F\) on the Fahrenheit scale and the temperature reading \(C\) on the Celsius scale is given by \(C=\frac{5}{9}(F-32)\). (a) Find the temperature at which the reading is the same on both scales. (b) When is the Fahrenheit reading twice the Celsius reading?

Exer. 65-66: Determine whether the point \(P\) is inside, outside, or on the circle with center \(C\) and radius \(r\). (a) \(P(3,8), \quad C(-2,-4), \quad r=13\) (b) \(P(-2,5), \quad C(3,7), \quad r=6\) (c) \(P(1,-2), \quad C(6,-7), \quad r=7\)

A certain paperback sells for \(\$ 12\). The author is paid royalties of \(10 \%\) on the first 10,000 copies sold, \(12.5 \%\) on the next 5000 copies, and \(15 \%\) on any additional copies. Find a piecewise-defined function \(R\) that specifies the total royalties if \(x\) copies are sold.

Vertical wind shear occurs when wind speed varies at different heights above the ground. Wind shear is of great importance to pilots during takeoffs and landings. If the wind speed is \(v_{1}\) at height \(h_{1}\) and \(v_{2}\) at height \(h_{2}\), then the average wind shear \(s\) is given by the slope formula $$ s=\frac{v_{2}-v_{1}}{h_{2}-h_{1}} . $$ If the wind speed at ground level is \(22 \mathrm{mi} / \mathrm{hr}\) and \(s\) has been determined to be \(0.07\), find the wind speed 185 feet above the ground.

Exer. 67-68: For the given circle, find (a) the \(x\)-intercepts and (b) the \(y\)-intercepts. $$ x^{2}+y^{2}-4 x-6 y+4=0 $$