(4 POINTS) Hang Time. One of the most exciting plays in basketball is the dunk shot. The amount of time T that passes from the moment a player leaves the ground, goes up, makes the shot, and arrives back on the ground is called the hang time. A function relating an athlete’s vertical leap V, in inches, to hang time T, in seconds, is given by

a.  Carl Landry, of Purdue University, has a hang time of about 0.835 sec. What is his vertical leap?

b.  The record for the greatest vertical leap in the NBA is held by Darryl Griffith of the Utah Jazz. It was 48 in. What was his hang time?

(a) V(0.835) = 48(0.835^2) = 33.47 sec
(b) T^2 = V/48
T^2 = 48/48 = 1
T = 1 sec

2.  (4 POINTS) Falling Distance. An object that is tossed downward with an initial speed (velocity) of will travel a distance of s meters, where and t is measured in seconds. Solve for t.

s = 4.9t^2 + v0 t
4.9t^2 + v0 t - s = 0
Using quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we get
t = [-v0 ± √{(v0)^2 - 4 * 4.9 * -s}] / (2 * 4.9)
t = [-v0 ± √{(v0)^2 + 19.6s}] / 9.8

3.  (4 points) Page Numbers. A student opens a literature book to two facing pages. The product of the page numbers is 812. Find the page numbers.

Let the page numbers be n and n + 1. Then we have
n(n + 1) = 812
n^2 + n - 812 = 0
n^2 + 29n - 28n - 812 = 0
n(n + 29) - 28(n + 29) = 0
(n + 29)(n - 28) = 0
n = {-29, 28}
The page numbers are 28 and 29.

4.  (4 POINTS) Architecture. An architect is designing the floors for a hotel. Each floor is to be rectangular and is allotted 720 ft of security piping around walls outside the rooms. What dimensions of the atrium will allow an atrium at the bottom to have maximum area?

Let the length be x and width be y
By data, 2x + 2y = 720
x + y = 360
y = 360 - x
Area = xy = x(360 - x) = 360x - x^2
The maximum area will occur when x = -b/2a = -360/(2 * -1) = 180
y = 360 - 180 = 180
The dimensions are 180 ft x 180 ft.

5.  (4 POINTS) Minimizing Cost. Aki’s Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by , where is in hundreds of dollars. How many bicycles should the shop build in order to minimize the average cost per bicycle?

Vertex of the graph of the given function occurs at x = -b/2a = -(-0.7) / (2 * 0.1) = 3.5
Minimum average cost will occur when x = 3.5 * 100 = 350 bicycles are built.