Question

deon throws a ball into the air. the height, h, of the ball, in meters, at time t seconds is modeled by the function $h(t) = -5t^2 + t + 4$.

a. write the equation that can be used to tell whether the ball reaches a height of 5m?

b. how long until the ball reaches 5 meters?

Explanation:

Part a

Step1: Set height to 5

We know the height function is \( h(t)= - 5t^{2}+t + 4 \). To find when the height is 5 meters, we set \( h(t) = 5 \).
So the equation is \( 5=-5t^{2}+t + 4 \).

Step2: Rearrange the equation

Subtract 5 from both sides to get it in standard quadratic form.
\( 0=-5t^{2}+t + 4 - 5 \)
Simplify to \( - 5t^{2}+t - 1=0 \) (or multiply both sides by - 1 to get \( 5t^{2}-t + 1 = 0 \))

Step1: Start with the equation from part a

We have the quadratic equation \( 5t^{2}-t + 1 = 0 \) (from rearranging \( 5=-5t^{2}+t + 4 \)). The quadratic formula is \( t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for an equation \( ax^{2}+bx + c = 0 \). Here, \( a = 5 \), \( b=-1 \), \( c = 1 \).

Step2: Calculate the discriminant

The discriminant \( D=b^{2}-4ac=(-1)^{2}-4\times5\times1=1 - 20=-19 \)

Step3: Analyze the discriminant

Since the discriminant \( D=-19<0 \), the quadratic equation has no real solutions. This means the ball never reaches a height of 5 meters.

Answer:

\( 5=-5t^{2}+t + 4 \) (or equivalent quadratic equation like \( 5t^{2}-t + 1 = 0 \))

Part b