Page 236 - How To Solve Word Problems In Calculus

The profit realized from the sale of the 100th unit is approximately
$54.The exact profit = P (100) − P (99) = 24,600 − 24,548 = $52.
6. Let x = supply (thousands of units)
p = price (dollars) per unit
dp dx
Given: = 1 Find: when p = 100
dt dt

√
2
x + 2x p − p = 25
2
x + 2xp 1/2 − p = 25

dx 1 −1/2 dp 1/2 dx dp
2x + 2 x p + (p ) − = 0
dt 2 dt dt dt
dx x dp √ dx dp
2x + √ + 2 p − = 0
dt p dt dt dt

dp
We know and we know p at the instant in question.We need
dt
to determine the corresponding value of x.

√
2
x + 2x p − p = 25
√
2
When p = 100, x + 2x 100 − 100 = 25
2
x + 20x − 100 = 25
2
x + 20x − 125 = 0
(x + 25)(x − 5) = 0

x = 5 (The value x =−25 is discarded.)

dp dx
We substitute x = 5, p = 100, = 1 and solve for :
dt dt

dx 5 √ dx
2(5) + √ (1) + 2 100 − 1 = 0
dt 100 dt
dx 1 dx
10 + + 20 − 1 = 0
dt 2 dt
223