Chapter 3

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. Tangent to both axes, center in the second quadrant, radius 4

Computer science functions Let the function \(\mathrm{CHR}\) be defined by \(\mathrm{CHR}(65)=\) "A", \(\mathrm{CHR}(66)=\) "B", ... \(\operatorname{CHR}(90)=" Z "\). Then let the function ORD be defined by \(\mathrm{ORD}(" \mathrm{~A} ")=65, \quad \mathrm{ORD}(" \mathrm{~B} ")=66, \ldots\), \(\operatorname{ORD}\left({ }^{\prime \prime Z "}\right)=90\). Find (a) \((\mathrm{CHR} \circ \mathrm{ORD})\left({ }^{\prime \prime} \mathrm{C}\right.\) ") (b) \(\mathrm{CHR}(\mathrm{ORD}(" A ")+3)\)

Find two real numbers whose difference is 40 and whose product is a minimum.

Exer. 41-44: Use the slope-intercept form to find the slope and \(y\)-intercept of the given line, and sketch its graph. $$ x-5 y=-15 $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=3 $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Tangent to both axes, center in the fourth quadrant, radius } 3 $$

A fire has started in a dry open field and is spreading in the form of a circle. If the radius of this circle increases at the rate of \(6 \mathrm{ft} / \mathrm{min}\), express the total fire area \(A\) as a function of time \(t\) (in minutes).

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=-\sqrt{36-x^{2}} $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Endpoints of a diameter } A(4,-3) \text { and } B(-2,7) $$

A spherical balloon is being inflated at a rate of \(\frac{9}{2} \pi \mathrm{ft}^{3} / \mathrm{min}\). Express its radius \(r\) as a function of time \(t\) (in minutes), assuming that \(r=0\) when \(t=0\).

A farmer wishes to put a fence around a rectangular field and then divide the field into three rectangular plots by placing two fences parallel to one of the sides. If the farmer can afford only 1000 yards of fencing, what dimensions will give the maximum rectangular area?

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=\sqrt{16-x^{2}} $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Endpoints of a diameter } A(-5,2) \text { and } B(3,6) $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}3 & \text { if } x \leq-1 \\ -2 & \text { if } x>-1\end{cases} $$

Dimensions of a sand pile The volume of a conical pile of sand is increasing at a rate of \(243 \pi \mathrm{ft}^{3} / \mathrm{min}\), and the height of the pile always equals the radius \(r\) of the base. Express \(r\) as a function of time \(t\) (in minutes), assuming that \(r=0\) when \(t=0\).

Flights of leaping animals typically have parabolic paths. The figure on the next page illustrates a frog jump superimposed on a coordinate plane. The length of the leap is 9 feet, and the maximum height off the ground is 3 feet. Find a standard equation for the path of the frog.

Exer. 47-48: If a line \(l\) has nonzero \(x\) - and \(y\)-intercepts \(a\) and \(b\), respectively, then its intercept form is $$ \frac{x}{a}+\frac{y}{b}=1 . $$ Find the intercept form for the given line. $$ 4 x-2 y=6 $$

Exer. 47-48: Simplify the difference quotient $$ \frac{f(2+h)-f(2)}{h} \text { if } h \neq 0 . $$ $$ f(x)=x^{2}-3 x $$

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}-4 x+6 y-36=0 $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}-1 & \text { if } x \text { is an integer } \\ -2 & \text { if } x \text { is not an integer }\end{cases} $$

The diagonal \(d\) of a cube is the distance between two opposite vertices. Express \(d\) as a function of the edge \(x\) of the cube. (Hint: First express the diagonal \(y\) of a face as a function of \(x\).)

In the \(1940 \mathrm{~s}\), the human cannonball stunt was performed regularly by Emmanuel Zacchini for The Ringling Brothers and Barnum \& Bailey Circus. The tip of the cannon rose 15 feet off the ground, and the total horizontal distance traveled was 175 feet. When the cannon is aimed at an angle of \(45^{\circ}\), an equation of the parabolic flight (see the figure) has the form \(y=a x^{2}+x+c\). (a) Use the given information to find an equation of the flight. (b) Find the maximum height attained by the human cannonball.

Exer. 47-48: If a line \(l\) has nonzero \(x\) - and \(y\)-intercepts \(a\) and \(b\), respectively, then its intercept form is $$ \frac{x}{a}+\frac{y}{b}=1 . $$ Find the intercept form for the given line. $$ x-3 y=-2 $$

Exer. 47-48: Simplify the difference quotient $$ \frac{f(2+h)-f(2)}{h} \text { if } h \neq 0 . $$ $$ f(x)=-2 x^{2}+3 $$

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}+8 x-10 y+37=0 $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}3 & \text { if } x<-2 \\ -x+1 & \text { if }|x| \leq 2 \\\ -3 & \text { if } x>2\end{cases} $$

Find an equation of the circle that has center \(C(3,-2)\) and is tangent to the line \(y=5\).

Exer. 49-50: Simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} \text { if } h \neq 0 . $$ $$ f(x)=x^{2}+5 $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}-2 x & \text { if } x<-1 \\ x^{2} & \text { if }-1 \leq x<1 \\ -2 & \text { if } x \geq 1\end{cases} $$

Find an equation of the line that is tangent to the circle \(x^{2}+y^{2}=25\) at the point \(P(3,4)\).

Exer. 49-50: Simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} \text { if } h \neq 0 . $$ $$ f(x)=1 / x^{2} $$

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}-10 x+18=0 $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}x+2 & \text { if } x \leq-1 \\ x^{3} & \text { if }|x|<1 \\\ -x+3 & \text { if } x \geq 1\end{cases} $$

A doorway has the shape of a parabolic arch and is 9 feet high at the center and 6 feet wide at the base. If a rectangular box 8 feet high must fit through the doorway, what is the maximum width the box can have?

The growth of a fetus more than 12 weeks old can be approximated by the formula \(L=1.53 t-6.7\), where \(L\) is the length (in centimeters) and \(t\) is the age (in weeks). Prenatal length can be determined by ultrasound. Approximate the age of a fetus whose length is 28 centimeters.

Exer. 51-52: Simplify the difference quotient \(\frac{f(x)-f(a)}{x-a}\) if \(x \neq a\). \(f(x)=\sqrt{x-3}\) (Hint: Rationalize the numerator.)

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ 2 x^{2}+2 y^{2}-12 x+4 y-15=0 $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}x-3 & \text { if } x \leq-2 \\ -x^{2} & \text { if }-2

A 100 -foot-long cable of diameter 4 inches is submerged in seawater. Because of corrosion, the surface area of the cable decreases at the rate of 750 in \(^{2}\) per year. Express the diameter \(d\) of the cable as a function of time \(t\) (in years). (Disregard corrosion at the ends of the cable.)

Assume a baseball hit at home plate follows a parabolic path having equation \(y=-\frac{3}{4000} x^{2}+\frac{3}{10} x+3\), where \(x\) and \(y\) are both measured in feet. (a) Find the maximum height of the baseball. (b) Does the baseball clear an 8-foot fence that is 385 feet from home plate?

Salinity of the ocean refers to the amount of dissolved material found in a sample of seawater. Salinity \(S\) can be estimated from the amount \(C\) of chlorine in seawater using \(S=0.03+1.805 C\), where \(S\) and \(C\) are measured by weight in parts per thousand. Approximate \(C\) if \(S\) is \(0.35\).

Exer. 51-52: Simplify the difference quotient \(\frac{f(x)-f(a)}{x-a}\) if \(x \neq a\). \(f(x)=x^{3}-2\)

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ 9 x^{2}+9 y^{2}+12 x-6 y+4=0 $$

Exer. 53-54: The symbol \(\llbracket x \rrbracket\) denotes values of the greatest integer function. Sketch the graph of \(f\). (a) \(f(x)=\llbracket x-3 \rrbracket\) (b) \(f(x)=\llbracket x \rrbracket-3\) (c) \(f(x)=2 \llbracket x \rrbracket\) (d) \(f(x)=\llbracket 2 x \rrbracket\) (e) \(f(x)=\llbracket-x \rrbracket\)

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

A company sells running shoes to dealers at a rate of \(\$ 40\) per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600 ), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company?

The expected weight \(W\) (in tons) of a humpback whale can be approximated from its length \(L\) (in feet) by using \(W=1.70 L-42.8\) for \(30 \leq L \leq 50\). (a) Estimate the weight of a 40-foot humpback whale. (b) If the error in estimating the length could be as large as 2 feet, what is the corresponding error for the weight estimate?

Exer. 53-54: If a linear function \(f\) satisfies the given conditions, find \(f(x)\). $$ f(-3)=1 \text { and } f(3)=2 $$

Exer. 47-56: Find the center and radius of the circle with the given equation. $$ x^{2}+y^{2}+4 x-2 y+5=0 $$

Exer. 53-54: The symbol \(\llbracket x \rrbracket\) denotes values of the greatest integer function. Sketch the graph of \(f\). (a) \(f(x)=\llbracket x+2 \rrbracket\) (b) \(f(x)=\llbracket x \rrbracket+2\) (c) \(f(x)=\frac{1}{2} \llbracket x \rrbracket\) (d) \(f(x)=\llbracket \frac{1}{2} x \rrbracket\) (e) \(f(x)=-\llbracket-x \rrbracket\)