A company makes and sells notebook computers. The company’s cost function in thousands of dollars is C(x) = 0.3x + 100, and the revenue function in thousands of dollars is R(x) = 0.75x, where x is the number of notebook computers.

(a) Evaluate and interpret C(100).
(b) Interpret the y-intercepts on the graphs of C and R.*
(c) Give the profit function P(x). P(x) = 0.45x - 100.
(d) How many computers need to be sold to make a profit?

To answer these questions, we will use the given cost and revenue functions and analyze their meanings and values.

(a) To evaluate C(100), we substitute 100 for x in the cost function C(x):

C(100) = 0.3(100) + 100
C(100) = 30 + 100
C(100) = 130

Interpretation: C(100) = 130 means that when 100 notebook computers are produced and sold, the cost incurred by the company is $130,000.

(b) The y-intercept on the graph of a function represents the value of the function when x is zero. Let's find the y-intercepts for both C(x) and R(x):

For the cost function C(x), we set x = 0:

C(0) = 0.3(0) + 100
C(0) = 100

Interpretation: The y-intercept for the cost function C(x) is 100, which implies that even when no notebook computers are produced and sold (x = 0), the company incurs a cost of $100,000.

For the revenue function R(x), we set x = 0:

R(0) = 0.75(0)
R(0) = 0

Interpretation: The y-intercept for the revenue function R(x) is 0, indicating that when no notebook computers are produced and sold (x = 0), the revenue is zero.

(c) The profit function P(x) can be obtained by subtracting the cost function C(x) from the revenue function R(x):

P(x) = R(x) - C(x)
P(x) = 0.75x - (0.3x + 100)
P(x) = 0.45x - 100

The profit function is given by P(x) = 0.45x - 100.

(d) To determine the number of computers needed to make a profit, we set the profit function P(x) greater than zero and solve for x:

0.45x - 100 > 0
0.45x > 100
x > 222.22 (approximately)

Interpretation: To make a profit, the company needs to sell more than 222 notebook computers. Since we cannot sell a fraction of a computer, the minimum number of computers required is 223.