Question

3.The monthly income from the sale of x units is given by: R(𝑥) = 12𝑥 − 0.01𝑥^2 Dollars Determine the number of units that must be sold each month to maximize income. How much is the maximum income?

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Maude

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67 Answers

To maximize the monthly income, we need to find the vertex of the parabola represented by the equation:

R(x) = 12x - 0.01x^2

The general form of a quadratic equation is:

R(x) = ax^2 + bx + c

For the given equation:

a = -0.01

b = 12

c = 0

The vertex form of a quadratic equation is given by:

x = -\frac{b}{2a}

Substitute a and b into the vertex form equation:

x = -\frac{12}{2(-0.01)}

x = -\frac{12}{-0.02}

x = 600

Thus, the number of units that must be sold each month to maximize income is:

x = 600

Next, substitute x = 600 back into the original equation to find the maximum income:

R(600) = 12(600) - 0.01(600)^2

R(600) = 7200 - 0.01 \times 360000

R(600) = 7200 - 3600

R(600) = 3600

Therefore, the maximum income is:

R(600) = 3600

The general form of a quadratic equation is:

For the given equation:

The vertex form of a quadratic equation is given by:

Substitute a and b into the vertex form equation:

Thus, the number of units that must be sold each month to maximize income is:

Next, substitute x = 600 back into the original equation to find the maximum income:

Therefore, the maximum income is:

R(600) = 3600

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