A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equation h=−16t2+60t+0

Question

ℎ
=
−
16
𝑡
2
+
60
𝑡
+
0
, where h is the height in feet and t is the time in seconds. Using this equation, what is the maximum range the ball will reach? Round to the nearest hundredth.(1 point)
feet

Answer 1

To find the maximum height (or range) the ball will reach using the equation $h=-16t^2+60t+0$, we need to determine the vertex of the parabola represented by this quadratic equation. The maximum height occurs at the vertex, which can be found using the formula for the time $t$ at which the vertex occurs:

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

where $a$ and $b$ are coefficients from the quadratic equation $h=a{t}^2+bt+c$.

In this case, the coefficients are:

• $a=-16$
• $b=60$
• $c=0$

Substituting $a$ and $b$ into the vertex formula:

$$t=-\frac{60}{2(-16)}=-\frac{60}{-32}=\frac{60}{32}=1.875,\text{seconds}$$

Next, we need to find the maximum height by substituting this value of $t$ back into the height equation:

$$h=-16(1.875{)}^2+60(1.875)+0$$

Calculating $(1.875{)}^2$:

$$(1.875{)}^2=3.515625$$

Now substitute this value back into the height equation:

$$h=-16(3.515625)+60(1.875)$$

Calculating each term:

$$h=-56.25+112.5$$

Now combine the terms:

$$h=-56.25+112.5=56.25$$

Thus, the maximum height (or range) the ball will reach is

$$\boxed{{\displaystyle 56.25}},\text{feet}$$

Rounding to the nearest hundredth, the answer remains $56.25$ feet.