To determine the initial velocity [tex]\( v \)[/tex] at which the quarterback must throw the football, we'll use the given information and the projectile motion formula:
[tex]\[
h(t) = -16t^2 + vt + h_0
\][/tex]
Given:
- The initial height [tex]\( h_0 = 6.5 \)[/tex] feet
- The final height [tex]\( h = 5 \)[/tex] feet when [tex]\( t = 3.5 \)[/tex] seconds
We need to find [tex]\( v \)[/tex].
Step 1: Substitute the known values into the projectile motion formula.
[tex]\[
h = -16t^2 + vt + h_0
\][/tex]
Substituting [tex]\( h = 5 \)[/tex], [tex]\( t = 3.5 \)[/tex], and [tex]\( h_0 = 6.5 \)[/tex]:
[tex]\[
5 = -16(3.5)^2 + v(3.5) + 6.5
\][/tex]
Step 2: Simplify the equation.
First, calculate [tex]\( 3.5^2 \)[/tex]:
[tex]\[
3.5^2 = 12.25
\][/tex]
Then multiply by -16:
[tex]\[
-16 \cdot 12.25 = -196
\][/tex]
So our equation becomes:
[tex]\[
5 = -196 + 3.5v + 6.5
\][/tex]
Now, combine the constants on the right side (-196 and 6.5):
[tex]\[
5 = -189.5 + 3.5v
\][/tex]
Step 3: Solve for [tex]\( v \)[/tex].
Isolate [tex]\( v \)[/tex] by adding 189.5 to both sides of the equation:
[tex]\[
5 + 189.5 = 3.5v
\][/tex]
[tex]\[
194.5 = 3.5v
\][/tex]
Next, divide both sides by 3.5:
[tex]\[
v = \frac{194.5}{3.5}
\][/tex]
Calculate the division:
[tex]\[
v \approx 55.5714285714286
\][/tex]
Step 4: Round [tex]\( v \)[/tex] to the nearest whole number.
[tex]\[
v \approx 56
\][/tex]
Thus, the initial velocity [tex]\( v \)[/tex], rounded to the nearest whole number, is:
[tex]\[
v \approx 56 \quad \text{feet per second}
\][/tex]
