Question

what is the maximum height of the T-shirt after being fired from the cannon? round to the nearest tenth. Will the fans sitting in the bleachers, 90 feet above the field, catch the t-shirt? Justify your response

Answer 1

$h(t)=-16t^2+88t+5$

Based on the given trajectory, the leading numerical coefficient is a negative number, hence, the graph of this function is a parabola opening downwards.

In a parabola opening downwards, the maximum value is the vertex of the parabola. So, in order to determine the maximum height, let's solve for the vertex of the parabola. The formula is:

$t=-\frac{b}{2a}$

where in the quadratic function is in the form of at² + bt + c = 0.

So, in the given trajectory function, a = -16, b = 88, and c = 5. Let's replace the variables in the formula above with their corresponding numerical value.

$t=-\frac{88}{2(-16)}$

Then, solve.

$t=-\frac{88}{(-32)}\Rightarrow t=2.75$

Therefore, at t = 2.75 seconds, the t-shirt reached its maximum height.

So, to determine the maximum height, replaced "t" in the trajectory function with 2.75.

$h(2.75)=-16(2.75^)^2+88(2.75)+5$$h(2.75)=-121+242+5$$h(2.75)=126$

Hence, after 2.75 seconds, the t-shirt reached its maximum height which is 126 feet.

Since the maximum height of the t-shirt is 126 feet, then yes, there is a chance that the fans sitting in the bleachers 90ft above the field will catch a T-shirt.