Answer:
The level of production that maximizes the profit is 90 units.
The maximum possible profit is 1520 dollars
Step-by-step explanation:
The profit is modeled by a quadratic equation, thus we can find its vertex x coordinate and we will find the number of units that maximizes the profit using the vertex formula
![x=-\cfrac {b}{2a}](https://tex.z-dn.net/?f=x%3D-%5Ccfrac%20%7Bb%7D%7B2a%7D)
For
![P(x) = ax^2+bx+c](https://tex.z-dn.net/?f=P%28x%29%20%3D%20ax%5E2%2Bbx%2Bc)
Then we can find the profit just replacing the number of units on the profit equation.
a) What level of production maximizes profit?
We can first write the profit equation in descending order.
![P(x) =-0.2x^2+36x-100](https://tex.z-dn.net/?f=P%28x%29%20%3D-0.2x%5E2%2B36x-100)
Thus we get for the coefficients a = -0.2 and b = 36.
Replacing them on the vertex formula
![x=-\cfrac{b}{2a}](https://tex.z-dn.net/?f=x%3D-%5Ccfrac%7Bb%7D%7B2a%7D)
We get
![x=-\cfrac{36}{2(-0.2)}](https://tex.z-dn.net/?f=x%3D-%5Ccfrac%7B36%7D%7B2%28-0.2%29%7D)
Simplifying we get
![\boxed{x=90\, units}](https://tex.z-dn.net/?f=%5Cboxed%7Bx%3D90%5C%2C%20units%7D)
The level of production that maximizes the profit is 90 units
b) What is the maximum possible profit?
We can replace the vertex x coordinate on the profit function, which is finding the value of the profit P(x) for x = 90 units.
![P(90)=-0.2(90)^2+36(90)-100](https://tex.z-dn.net/?f=P%2890%29%3D-0.2%2890%29%5E2%2B36%2890%29-100)
Simplifying we get
![\boxed{P(90) = \$\, 1520}](https://tex.z-dn.net/?f=%5Cboxed%7BP%2890%29%20%3D%20%5C%24%5C%2C%201520%7D)
The maximum possible profit is 1520 dollars
A - 1.2b = - 3 ⇒ eq. 1
0.2b + 0.6a = 12 ⇒ eq. 2
a - 1.2b = - 3
a = 1.2b - 3 ⇒ eq. 3
Substitute eq. 3 to eq. 2
0.2b + 0.6(1.2b - 3) = 12
0.2b + .72b - 1.8 = 12
Combine like terms
0.92b = 12 + 1.8
0.92b = 13.8 ⇒ divide both sides by 0.92 to get the value of b
<span>b = 15
</span>Substitute the value of b to eq. 3
a = 1.2b - 3
a = 1.2(15) - 3
<span>a = 15
</span>to check: substitute the values of a and b to:
eq. 1
a - 1.2b = - 3
15 - (1.2)(15) = -3
15 - 18 = - 3
- 3 = -3 ⇒ correct!
eq. 2
0.2b + 0.6a = 12
(0.2)(15) + (0.6)(15) = 12
3 + 9 = 12
12 = 12 ⇒ correct!
Amount used = $25 - $23.35 = $1.65
Number of minutes used = $1.65 ÷ $0.11 = 15 mins
Answer: 15 mins
The third diagram is the correct answer.
Answer: 4.02 c
Step-by-step explanation:
so the area of a circle is equal to A=pi multiplied by the radius to the second power. And since that’s half a circle it would make sense for the area to be half of the full circle. The radius (half of the diameter and the area from the center of the circle to the outer edge.) is shown as 1.6ft, 1.6^2 is 2.56 and multiplied by pi that is 8.04 half of that is 4.02.