Solve the given problem.

A frog leaps from a stump three feet high and lands four feet from the base of the stump. We can consider the initial position of the frog to be at $(0,3)$ and its landing position to be at $(4,0)$.

It is determined that the height $h$ in feet of the frog as a function of its distance $x$ from the base of the stump is given by

$$h(x)=-0.5 x^2+1.25 x+3 .$$

(a) How high was the frog when its horizontal distance $x$ from the base of the stump was $2$ feet?

(b) What was the horizontal distance from the base of the stump when the frog was $3.25$ feet above the ground?

(c) At what horizontal distance from the base of the stump did the frog reach its highest point?

(d) What was the maximum height reached by the frog?

Find the $\left[\mathrm{H}_3 \mathrm{O}^{+}\right]$for each substance with the given $p H$. See the example in this section.

soda pop, $2.7$

Find the pH for each substance with the given hydronium ion $\left[\mathrm{H}_3 \mathrm{O}^{+}\right]$concentration. Refer to the earlier example.

Crackers, $3.9 \times 10^{-9}$

Describe how to transform the graph of an appropriate monomial function

$$f(x)=x^n$$

into the graph of the given polynomial function. Sketch the transformed graph by hand and support your answer with a grapher. Compute the location of the y-intercept as a check on the transformed graph.

$$g(x)=-2(x+2)^4-3$$

Graph this polynomial function by hand, as shown in the previous section. Then solve each equation or inequality. In part (a), state if the multiplicity of a solution is greater than one.

$$\begin{aligned}P(x)&=x^3-3 x^2-6 x+8\\ &=(x-4)(x-1)(x+2)\end{aligned}$$

(a) $P(x)=0$

(b) $P(x)<0$

(c) $P(x)>0$

When an object, such as a bowling ball, is dropped near Earth's surface, the distance d(t) (in feet) that the object falls in t seconds is given by $d(t)=16 t^{2}$. Its velocity (in feet per second) after 2 seconds is given by $\lim _{t \rightarrow 2} \frac{d(t)-d(2)}{t-2}$. Evaluate this limit algebraically to find the velocity of the bowling ball after 2 seconds.

For the given situation, if $x$ represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analytically how many items must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically.

The fixed cost is $\$ 180$, the cost to produce an item is $\$ 11$, and the selling price of the item is $\$ 20$.

Suppose that speeding fines are determined by $y=10(x-65)+50, x>65$, where y is the cost in dollars of the fine if a person is caught driving x miles per hour.

(a) How much is the fine for driving 76 mph?

(b) While balancing the checkbook, Johnny found a check that his wife Gwen had written to the Department of Motor Vehicles for a speeding fine. The check was written for $100. How fast was Gwen driving?

(c) At what whole-number speed are tickets first given?

(d) For what speeds is the fine greater than$200?

The exercise given below involves octane rating of gasoline, a measure of its antiknock qualities. In one measure of octane, a standard fuel is made with only two ingredients: heptane and isooctane. For this fuel, the octane rating is the percent of isooctane. An actual gasoline blend is then compared with a standard fuel. For example, a gasoline with an octane rating of 98 has the same antiknock properties as a standard fuel that is 98% isooctane.

How many gallons of 92-octane and 98-octane gasoline should be mixed together to provide 120 gallons of 96-octane gasoline?

Solve the problem analytically, and support your solution graphically.

The length of a rectangular mailing label is 3 centimeters less than twice the width. The perimeter is 54 centimeters. Find its dimensions.

Solve each problem analytically, and support your solution graphically.

The length of a rectangular mailing label is 3 centimeters less than twice the width. The perimeter is 54 centimeters. Find its dimensions.

piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 2-in. squares from each corner and folding up the sides. Let x represent the width (in inches) of the original piece of cardboard. (a) Represent the length of the original piece of cardboard in terms of x. (b) What will be the dimensions of the bottom rectangular base of the box? Give the restrictions on x. (c) Determine a function V that represents the volume of the box in terms of x. $(d) For what dimensions of the bottom of the box will the volume be 320$ $in.^3$? (e) Find the values of x if such a box is to have a volume between 400 and 500 $in.^3$.

A farmer has 1000 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enclosed by the fence?

How is the concept of technology, as defined with the aggregate production function, different from our everyday use of the word?