Chapter 3

An electric company charges its customers \(\$ 0.0577\) per kilowatt-hour \((\mathrm{kWh})\) for the first \(1000 \mathrm{kWh}\) used, \(\$ 0.0532\) for the next \(4000 \mathrm{kWh}\), and \(\$ 0.0511\) for any \(\mathrm{kWh}\) over 5000 . Find a piecewise-defined function \(C\) for a customer's bill of \(x \mathrm{kWh}\).

An aquarium of height \(1.5\) feet is to have a volume of \(6 \mathrm{ft}^{3}\). Let \(x\) denote the length of the base and \(y\) the width (see the figure). (a) Express \(y\) as a function of \(x\). (b) Express the total number \(S\) of square feet of glass needed as a function of \(x\).

Exer. 67-68: For the given circle, find (a) the \(x\)-intercepts and (b) the \(y\)-intercepts. $$ x^{2}+y^{2}-10 x+4 y+13=0 $$

There are two car rental options available for a four-day trip. Option I is \(\$ 45\) per day, with 200 free miles and \(\$ 0.40\) per mile for each additional mile. Option II is \(\$ 58.75\) per day, with a charge of \(\$ 0.25\) per mile. (a) Determine the cost of a 500 -mile trip for both options. (b) Model the data with a cost function for each fourday option. (c) Determine the mileages at which each option is preferable.

Exer. 69-70: The given points were found using empirical methods. Determine whether they lie on the same line \(y=a x+b\), and if so, find the values of \(a\) and \(b\). $$ \begin{array}{ll} A(-1.3,-1.3598), & B(-0.55,-1.11905) \\ C(1.2,-0.5573), & D(3.25,0.10075) \end{array} $$

A city council is proposing a new skyline ordinance. It would require the setback \(S\) for any building from a residence to be a minimum of 100 feet, plus an additional 6 feet for each foot of height above 25 feet. Find a linear function for \(S\) in terms of \(h\).

Find an equation of the circle that is concentric (has the same center ) with \(x^{2}+y^{2}+4 x-6 y+4=0\) and passes through \(P(2,6)\).

Cars are crossing a bridge that is 1 mile long. Each car is 12 feet long and is required to stay a distance of at least \(d\) feet from the car in front of it (see figure). (a) Show that the largest number of cars that can be on the bridge at one time is \(\llbracket 5280 /(12+d) \rrbracket\), where \(\llbracket \rrbracket\) denotes the greatest integer function. (b) If the velocity of each car is \(v \mathrm{mi} / \mathrm{hr}\), show that the maximum traffic flow rate \(F\) (in cars/hr) is given by \(F=\llbracket 5280 v /(12+d) \rrbracket\).

Exer. 69-70: The given points were found using empirical methods. Determine whether they lie on the same line \(y=a x+b\), and if so, find the values of \(a\) and \(b\). $$ \begin{array}{ll} A(-0.22,1.6968), & B(-0.12,1.6528) \\ C(1.3,1.028) & D(1.45,0.862) \end{array} $$

A proposed energy tax \(T\) on gasoline, which would affect the cost of driving a vehicle, is to be computed by multiplying the number \(x\) of gallons of gasoline that you buy by 125,000 (the number of BTUs per gallon of gasoline) and then multiplying the total BTUs by the tax \(-34.2\) cents per million BTUs. Find a linear function for \(T\) in terms of \(x\).

The signal from a radio station has a circular range of 50 miles. A second radio station, located 100 miles east and 80 miles north of the first station, has a range of 80 miles. Are there locations where signals can be received from both radio stations? Explain your answer.

For children between ages 6 and 10 , height \(y\) (in inches) is frequently a linear function of age \(t\) (in years). The height of a certain child is 48 inches at age 6 and \(50.5\) inches at age 7 . (a) Express \(y\) as a function of \(t\). (b) Sketch the line in part (a), and interpret the slope. (c) Predict the height of the child at age 10 .

A circle \(C_{1}\) of radius 5 has its center at the origin. Inside this circle there is a first-quadrant circle \(C_{2}\) of radius 2 that is tangent to \(C_{1}\). The \(y\)-coordinate of the center of \(C_{2}\) is 2 . Find the \(x\)-coordinate of the center of \(C_{2}\).

It has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of 40,000 \(\mathrm{km}^{2}\) in 40 days. Assuming that the area covered by the radioactive substance is a linear function of time \(t\) and is always circular in shape, express the radius \(r\) of the contamination as a function of \(t\).

A circle \(C_{1}\) of radius 5 has its center at the origin. Outside this circle is a first-quadrant circle \(C_{2}\) of radius 2 that is tangent to \(C_{1}\). The \(y\)-coordinate of the center of \(C_{2}\) is 3 . Find the \(x\)-coordinate of the center of \(C_{2}\).

A hot-air balloon is released at 1:00 P.M. and rises vertically at a rate of \(2 \mathrm{~m} / \mathrm{sec}\). An observation point is situated 100 meters from a point on the ground directly below the balloon (see the figure). If \(t\) denotes the time (in seconds) after 1:00 P.M., express the distance \(d\) between the balloon and the observation point as a function of \(t\).

From an exterior point \(P\) that is \(h\) units from a circle of radius \(r\), a tangent line is drawn to the circle (see the figure). Let \(y\) denote the distance from the point \(P\) to the point of tangency \(T\). (a) Express \(y\) as a function of \(h\). (Hint: If \(C\) is the center of the circle, then \(P T\) is perpendicular to \(C T\).) (b) If \(r\) is the radius of Earth and \(h\) is the altitude of a space shuttle, then \(y\) is the maximum distance to Earth that an astronaut can see from the shuttle. In particular, if \(h=200 \mathrm{mi}\) and \(r=4000 \mathrm{mi}\), approximate \(y\).

Exer. 73-76: Express, in interval form, the \(x\)-values such that \(y_{1}

The figure illustrates the apparatus for a tightrope walker. Two poles are set 50 feet apart, but the point of attachment \(P\) for the rope is yet to be determined. (a) Express the length \(L\) of the rope as a function of the distance \(x\) from \(P\) to the ground. (b) If the total walk is to be 75 feet, determine the distance from \(P\) to the ground.

A man in a rowboat that is 2 miles from the nearest point \(A\) on a straight shoreline wishes to reach a house located at a point \(B\) that is 6 miles farther down the shoreline (see the figure). He plans to row to a point \(P\) that is between \(A\) and \(B\) and is \(x\) miles from the house, and then he will walk the remainder of the distance. Suppose he can row at a rate of \(3 \mathrm{mi} / \mathrm{hr}\) and can walk at a rate of \(5 \mathrm{mi} / \mathrm{hr}\). If \(T\) is the total time required to reach the house, express \(T\) as a function of \(x\).