Questions: T(p)=-150(p-4)^2+1500 (c) (4 points) Calculate the p-intercepts of the parabola. (d) (2 points) Explain what your p-intercepts mean in the context of the problem. Include units. (e) (4 points) Calculate the average rate of change of T(p) on the interval [2,4]. (f) (2 points) Explain what your average rate of change means in the context of the problem. Include units.

Solution Steps

To solve the given questions, we need to follow these steps:

(c) To find the p-intercepts of the parabola, we need to set \( T(p) = 0 \) and solve for \( p \).

(e) To calculate the average rate of change of \( T(p) \) on the interval \([2, 4]\), we use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{T(4) - T(2)}{4 - 2} \]

To find the p-intercepts of the parabola defined by \( T(p) = -150(p-4)^{2} + 1500 \), we set \( T(p) = 0 \):

\[ -150(p-4)^{2} + 1500 = 0 \]

Solving for \( p \), we find:

\[ (p - 4)^{2} = 10 \implies p - 4 = \pm \sqrt{10} \]

Thus, the p-intercepts are:

\[ p = 4 - \sqrt{10} \quad \text{and} \quad p = 4 + \sqrt{10} \]

The average rate of change of \( T(p) \) on the interval \([2, 4]\) is given by:

\[ \text{Average Rate of Change} = \frac{T(4) - T(2)}{4 - 2} \]

Calculating \( T(2) \) and \( T(4) \):

\[ T(2) = 900 \quad \text{and} \quad T(4) = 1500 \]

Substituting these values into the formula:

\[ \text{Average Rate of Change} = \frac{1500 - 900}{2} = \frac{600}{2} = 300 \]

Final Answer