Questions: The height y (in feet) of a baseball t seconds after it is hit can be modeled by the function y=-5t^2+20t+60. When will the ball first reach 75 feet above ground?

The height y (in feet) of a baseball t seconds after it is hit can be modeled by the function y=-5t^2+20t+60. When will the ball first reach 75 feet above ground?

Transcript text: 13. The height $y$ (in feet) of a baseball $t$ seconds after it is hit can be modeled by the function $y=-5 t^{2}+20 t+60$. When will the ball first reach 75 feet above ground?

Solution

Solution Steps

Step 1: Set Up the Equation

To determine when the baseball first reaches a height of 75 feet, we start with the height function given by

\[ y = -5t^{2} + 20t + 60 \]

We set this equal to 75 feet:

\[ -5t^{2} + 20t + 60 = 75 \]

Step 2: Simplify the Equation

Next, we simplify the equation by moving 75 to the left side:

\[ -5t^{2} + 20t + 60 - 75 = 0 \]

This simplifies to:

\[ -5t^{2} + 20t - 15 = 0 \]

Step 3: Factor the Polynomial

We factor the polynomial $ -5t^{2} + 20t - 15 $. The factored form is:

\[ -5(t - 3)(t - 1) = 0 \]

Step 4: Solve for $ t $

Setting each factor equal to zero gives us the solutions:

\[ t - 3 = 0 \quad \Rightarrow \quad t = 3 \] \[ t - 1 = 0 \quad \Rightarrow \quad t = 1 \]

Step 5: Identify the First Time

The solutions for $ t $ are $ t = 1 $ seconds and $ t = 3 $ seconds. The first time the baseball reaches a height of 75 feet is:

\[ t = 1 \text{ seconds} \]

Final Answer

$\boxed{t = 1}$

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!