C(x) = $70(x) - $1000

If Jill can sell her lawnmowers for $150 each,
b) Find the revenue function R(x)
c) Find the manufactures weekly profit function P(x)
d) how many lawnmowers must the producer sell per week in order to show profit?

To find the revenue function, we need to multiply the number of lawnmowers sold, x, by the selling price per lawnmower, which is $150. So, the revenue function R(x) can be calculated as:

R(x) = Selling Price per Lawmower * Number of Lawmowers Sold
= $150 * x

Therefore, the revenue function R(x) is R(x) = $150x.

To find the manufacturer's weekly profit function, we need to subtract the cost function, C(x), from the revenue function, R(x). So, the profit function P(x) can be calculated as:

P(x) = R(x) - C(x)
= ($150x) - ($70x - $1000)
= $150x - $70x + $1000
= $80x + $1000

Therefore, the manufacturer's weekly profit function P(x) is P(x) = $80x + $1000.

To determine when the producer will show a profit, we need to find the value of x (number of lawnmowers sold per week) that makes the profit function P(x) positive. In other words, we need to solve the inequality: P(x) > 0.

$80x + $1000 > 0

To solve this inequality, we subtract $1000 from both sides of the inequality:

$80x > -$1000

Then, divide both sides of the inequality by $80:

x > -$1000 / $80

Simplifying:

x > -12.5

Since we cannot sell a fraction of a lawnmower, we round up to the nearest whole number:

x > 13

Therefore, the producer must sell at least 14 lawnmowers per week in order to show a profit.