Page 238 - How To Solve Word Problems In Calculus

8. For convenience, we will let n represent the number of cars in units
of 1000 and t, the toll, in dollars.If x represents the number of
dimes to be charged in excess of $2.00, the toll will be
t = 2.00 + 0.10x.Since the number of cars will decrease by 10,000
for each unit increment in x, the daily number of cars using the road
will be n = 300 − 10x thousand.The total revenue
R(x) = n · t

= (300 − 10x)(2.00 + 0.10x)

= 600 + 10x − x 2

R (x) = 10 − 2x
0 = 10 − 2x
x = 5

The toll should be increased by 50 cents (5 × 10 cents) to $2.50.
9. The profit function P (x) = R(x) − C (x).Since the company sells
the cameras at $60 each, R(x) = 60x.It follows that
3
2
P (x) = 60x − (40 + 4x − 1.6x + 0.1x )
2
= 60x − 40 − 4x + 1.6x − 0.1x 3
2
= 56x − 40 + 1.6x − 0.1x 3 0 ≤ x ≤ 30

P (x) = 56 + 3.2x − 0.3x 2
0 = 56 + 3.2x − 0.3x 2
We solve this equation by using the quadratic formula. a =−0.3,
b = 3.2, c = 56.
√
2
−b ± b − 4ac
x =
2a

2
−3.2 ± 3.2 − 4(−0.3)(56)
=
−0.6
√
−3.2 ± 77.44
=
−0.6
−3.2 ± 8.8
=
−0.6
x = 20 x =−9.33
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