91Ó°ÊÓ

Find the rule of the quadratic function whose graph passes through the given points (one of which is the vertex. $$(0,6),(-1,7),(2,10)$$

Find the number \(c\) such that the vertex of the parabola \(y=x^{2}+8 x+c\) lies on the \(x\) -axis.

Find two positive numbers whose sum is 111 and with the sum of their squares as small as possible.

Use the Factor Theorem and a calculator to factor the polynomial, as in Example 7. $$f(x)=6 x^{3}-7 x^{2}-89 x+140$$

A box with a square base and a volume of 1000 cubic inches is to be constructed. The material for the top and bottom of the box costs \(\$ 3\) per 100 square inches, and the material for the sides costs \(\$ 1.25\) per 100 square inches. (a) If \(x\) is the length of a side of the base, express the cost of constructing the box as a function of \(x .\) (b) If the side of the base must be at least 6 inches long, for what value of \(x\) will the cost of the box be \(\$ 7.50 ?\)

A field bounded on one side by a river is to be fenced on three sides so as to form a rectangular enclosure. If 200 feet of fencing is to be used, what dimensions will yield an enclosure of the largest possible area?

A rectangular garden with an area of 200 square meters is to be located next to a building and fenced on three sides, with the building acting as a fence on the fourth side. (a) If the side of the garden parallel to the building has length \(x\) meters, express the amount of fencing needed as a function of \(x\). (b) For what values of \(x\) will less than 60 meters of fencing be needed? (c) What value of \(x\) will result in the least possible amount of fencing being used? What are the dimensions of the garden in this case?

A certain company has fixed costs of \(\$ 40,000\) and variable costs of \(\$ 2.60\) per unit. (a) Let \(x\) be the number of units produced. Find the rule of the average cost function. [The average cost is the cost of the units divided by the number of units.] (b) Graph the average cost function in a window with \(0 \leq x \leq 100,000\) and \(0 \leq y \leq 20\). (c) Find the horizontal asymptote of the average cost function. Explain what the asymptote means in this situation. [How low can the average cost possibly be?]

You will need the formula for the height \(h\) of an object above the ground at time \(t\) seconds: $$h=-16 t^{2}+v_{0} t+h_{0}$$ this formula was explained on page 249 A toy rocket is fired straight up from ground level with an initial velocity of 80 feet per second. During what time interval will it be at least 64 feet above the ground?