Answers:
a. The company needs to sell 100 units to reach the maxmum profit.
b. The company's maximum profit is $5,000.
Solution:
a. How many unit does the company need to sell to reach the maximum profit?
p(x)=-0.5x^2+100x
This is a quadratic equation, and its graph is a parabola vertical (because the veriable "x" is square). Comparing with the general form:
p(x)=ax^2+bx+c; a=-0.5, b=100, c=0
a=-0.5<0 (negative), then the parabola opens downward, and it has a maximimun value (maximum profit) at its vertex.
We can find the abscissa of the vertex (units that the company needs to sell to reach the maximum profit using the following formula:
x=-b/(2a)
Replacing b by 100 and a by -0.5 in the formula above:
x=-100/[2(-0.5)]
x=-100/(-1)
x=100
The company needs to sell 100 units to reach the maximum profit.
b. What's the company's maximum profit?
To determine the maximum profit we substitute the value of "x" obrained in part "a" in the quadratic equation:
x=100→p(100)=-0.5(100)^2+100(100)
p(100)=-0.5(10,000)+10,000
p(100)=-5,000+10,000
p(100)=5,000
The company's maximum profit is $5,000.
so in terms of x we write