1. If a rock is thrown upward on the planet Mars with a velocity of 15 m/s, its

Question

1. If a rock is thrown upward on the planet Mars with a velocity of 15 m/s, its height (in meters) after t seconds is given by H = 15t 1.86t2.

(a) Find the velocity of the rock after one second.
m/s

(b) Find the velocity of the rock when t = a.
m/s

(c) When will the rock hit the surface? (Round your answer to one decimal place.)
t =  s

(d) With what velocity will the rock hit the surface?
m/s

2)The cost (in dollars) of producing x units of a certain commodity is given below.

C(x) = 6000 + 13x + 0.1x2

(a) Find the average rate of change of C with respect to x when the production is changed from x = 100 to the given value.

(i) x = 104
$  per unit

(ii) x = 101
$  per unit

(b) Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the marginal cost.)
$  per unit

Explanation / Answer

1. a)

H = 15t - 1.86t²

velocity is the derivative of H

V = H ' = 15 - 3.72t

after 1 sec that's 15 - 3.72(1) = 11.28m

b)
V = 15 - 3.72a

c)
The rock will hit the surface when H = 0

H = 15t - 1.86t² = 0
t(15 - 1.86t) = 0
t = 0 (when it was thrown)
or
15 - 1.86t = 0
1.86t = 15
t = 8.06 sec

d)
want velocity when t = 8.06 sec

V = H ' = 15 - 3.72*8.06

V = - 14.98 m/s

2)

C(104) = 6000 + 8*104 + 0.1*104^2 = 7913.6
C(100) = 6000 + 8*100 + 0.1*100^2 = 7800

average rate of change = 113.6 / 4 = 28.4


dC/dx = 8 + 0.2x

instantaneous rate of change when x is 100
= 8 + 0.2*100
= 28

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