Question

7: The average income, I, in dollars, of a lawyer with an age of x years is modeled with the following function: I = -425x^2 + 45,500x – 650,000
What is the youngest age for which the average income of a lawyer is $450,000?

Round answer to nearest year. Answer: _____

8: The average annual income, I, in dollars of a lawyer with an age of x years is modeled with the following function:

I = 425x^2 + 45,500x – 650,000

According to this model, what is the maximum average annual income in dollars, a lawyer can earn?

Round to the nearest whole dollar.

Answer: ______

Answer 1

Answer on Question #55824 – Math – Algebra

Task 7. The average income, I, in dollars, of a lawyer with an age of x years is modeled with the following function: $I = -425x^2 + 45,500x - 650,000$. What is the youngest age for which the average income of a lawyer is $450,000? Round answer to nearest year.

Solution

Solve the following equation:

$$-425x^2 + 45,500x - 650,000 = 450,000 \Leftrightarrow -425x^2 + 45,500x - 1100,000 = 0$$

Divide both sides of the equation by -25:

$$17x^2 - 1820x + 44,000 = 0$$

Using formulas for quadratic equation, we get:

$$x_1 = \frac{1820 + \sqrt{1820^2 - 4 * 17 * 44,000}}{2 * 17} = \frac{1820 + \sqrt{320400}}{34} \approx 70$$ $$x_2 = \frac{1820 - \sqrt{1820^2 - 4 * 17 * 44,000}}{2 * 17} = \frac{1820 - \sqrt{320400}}{34} \approx 37$$

Answer: 37 years.

Task 8. The average annual income, I, in dollars of a lawyer with an age of x years is modeled with the following function: $I = -425x^2 + 45,500x - 650,000$. According to this model, what is the maximum average annual income in dollars, a lawyer can earn? Round to the nearest whole dollar.

Solution

Consider function:

$$I = f(x) = -425x^2 + 45,500x - 650,000$$

This is a parabola. Using formula for the vertex of a parabola, we get:

$$x_{\text{vertex}} = -\frac{b}{2a} = -\frac{45500}{2 \cdot (-425)} \approx 53.5$$

Then, find $y_{\text{vertex}}$:

$$y_{\text{vertex}} = f(x_{\text{vertex}}) = -\frac{D}{4a} = -\frac{45,500^2 - 4 \cdot (-425) \cdot (-650,000)}{4 \cdot (-425)} \approx 567,794$$

Answer: $567,794.

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