To solve the problem, we need to evaluate the piecewise function \( f(t) \) at \( t = -10 \). Let's carefully analyze the given piecewise function and determine which part of the function applies when \( t = -10 \).
The piecewise function is defined as:
\[
f(t) =
\begin{cases}
t^2 - 5t & \text{if } t \leq -10 \\
t + 19 & \text{if } -10 < t < -2 \\
\frac{t^3}{t+9} & \text{if } t \geq -2
\end{cases}
\]
Step 1: Identify the relevant case for \( t = -10 \)
We need to check which condition \( t = -10 \) satisfies:
- The first case applies when \( t \leq -10 \).
- The second case applies when \( -10 < t < -2 \).
- The third case applies when \( t \geq -2 \).
Since \( t = -10 \) satisfies \( t \leq -10 \), we use the first case of the piecewise function:
\[
f(t) = t^2 - 5t \quad \text{for } t \leq -10
\]
Step 2: Substitute \( t = -10 \) into the first case
Using the formula \( f(t) = t^2 - 5t \):
\[
f(-10) = (-10)^2 - 5(-10)
\]
Step 3: Simplify the expression
Calculate each term:
\[
(-10)^2 = 100
\]
\[
-5(-10) = 50
\]
Add these results together:
\[
f(-10) = 100 + 50 = 150
\]
Final Answer:
\[
\boxed{150}
\]
Parent Tip: Review the logic above to help your child master the concept of evaluating piecewise functions worksheet.
