Math  /  Algebra

Question13. A company manufactures and sells a product. The revenue, in dollars, from selling xx units is given by the equation: R(x)=2x2+40xR(x)=-2 x^{2}+40 x
Find the number of units that must be sold to achieve a revenue of $200\$ 200.

Studdy Solution

STEP 1

1. The revenue function is quadratic, indicating a parabolic relationship between units sold and revenue.
2. We need to solve for x x when the revenue R(x) R(x) is set to \$200.
3. The quadratic equation may have two solutions, one solution, or no real solutions.

STEP 2

1. Set the revenue equation equal to 200.
2. Rearrange the equation into standard quadratic form.
3. Solve the quadratic equation using the quadratic formula.
4. Determine the valid solution(s) for the number of units.

STEP 3

Set the revenue equation equal to 200:
2x2+40x=200 -2x^2 + 40x = 200

STEP 4

Rearrange the equation into standard quadratic form ax2+bx+c=0 ax^2 + bx + c = 0 :
2x2+40x200=0 -2x^2 + 40x - 200 = 0

STEP 5

Use the quadratic formula to solve for x x . The quadratic formula is given by:
x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Identify a=2 a = -2 , b=40 b = 40 , and c=200 c = -200 .

STEP 6

Calculate the discriminant b24ac b^2 - 4ac :
b24ac=4024(2)(200) b^2 - 4ac = 40^2 - 4(-2)(-200) =16001600 = 1600 - 1600 =0 = 0

STEP 7

Since the discriminant is 0 0 , there is exactly one real solution. Substitute into the quadratic formula:
x=40±02(2) x = \frac{-40 \pm \sqrt{0}}{2(-2)} x=404 x = \frac{-40}{-4} x=10 x = 10

STEP 8

Verify that the solution x=10 x = 10 is valid in the context of the problem. Since it is a positive integer, it is a valid number of units.
The number of units that must be sold to achieve a revenue of \$200 is:
10 \boxed{10}

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