Answer:
a)
i) To calculate the profit if 0 cakes are made per day, we can substitute x = 0 into the function:
P(x) = -x²+36x - 40
P(0) = -0² + 36(0) - 40 = -40 dollars
So, the profit from making 0 cakes per day is -40 dollars.
ii) To calculate the profit if 20 cakes are made per day, we can substitute x = 20 into the function:
P(x) = -x²+36x - 40
P(20) = -20² + 36(20) - 40 = 760 dollars
So, the profit from making 20 cakes per day is 760 dollars.
b) To find out how many cakes need to be made per day for the profit to be $270, we need to solve the equation P(x) = 270.
We can substitute the function P(x) into this equation:
-x²+36x - 40 = 270
Next we can solve this equation for x by using any method you prefer ( factoring, Quadratic formula, etc)
x = (36 +/- √(36² -4*(-40)(-1)))/ 2(-1)
x = (36 +/- √(1296 + 320))/ -2
x = (36 +/- √1616)/ -2
x = (36 +/- 40)/ -2
x = -2 +/- 20
x = 18 or -22
The number of cakes need to be made per day for the profit to be $270 is 18 or -22, but -22 is not a valid solution because the number of cakes must be a non-negative value, so the number of cakes need to be 18