Answer
a) $D_{\theta=40^{\circ}}=67.00ft$; $D_{\theta=42^{\circ}}=67.14ft$; $D_{\theta=45^{\circ}}=66.84ft$; Distance $D$ increases as $\theta$ increase until the angle reaches $45^{\circ}$, when this happens this is the maximum range of projectile motion but it decreases the horizontal distance from the shot.
b) $D_{v=43ft/s}=64.40ft$; $D_{v=44ft/s}=67.14ft$; $D_{v=45ft/s}=69.93ft$; Distance $D$ increases as $v$ increases.
c) From the answers to $a$ and $b$, we see that the most effective way to increase distance $D$ is by increasing $v$. Therefore the shot-putter should increase the initial velocity to improve the performance.
Work Step by Step
Solving $a$,
Using the distance of a shot put formula and substituting the variables.
When $\theta=40^{\circ}$,
$D=\frac{44^{2}\sin(40)\cos(40)+44\cos(40)\sqrt ((44\sin(40))^{2}+64*7)}{32}$
$D=66.9994ft\approx67.00ft$ (rounded to the nearest hundredth)
When $\theta=42^{\circ}$,
$D=\frac{44^{2}\sin(42)\cos(42)+44\cos(42)\sqrt ((44\sin(42))^{2}+64*7)}{32}$
$D=67.1360ft\approx67.14ft$ (rounded to the nearest hundredth)
When $\theta=45^{\circ}$,
$D=\frac{44^{2}\sin(45)\cos(45)+44\cos(45)\sqrt ((44\sin(45))^{2}+64*7)}{32}$
$D=66.8364ft\approx66.84ft$ (rounded to the nearest hundredth)
From this data we can see that the distance $D$ increases as $\theta$ increase until the angle reaches $45^{\circ}$, when this happens this is the maximum range of projectile motion but it decreases the horizontal distance from the shot.
Solving $b$,
Using the distance of a shot put formula and substituting the variables.
When $v=43 ft/s$,
$D=\frac{43^{2}\sin(42)\cos(42)+43\cos(42)\sqrt ((43\sin(42))^{2}+64*7)}{32}$
$D=64.4016ft\approx64.40ft$ (rounded to the nearest hundredth)
When $v=44ft/s$,
$D=\frac{44^{2}\sin(42)\cos(42)+44\cos(42)\sqrt ((44\sin(42))^{2}+64*7)}{32}$
$D=67.1360ft\approx67.14ft$ (rounded to the nearest hundredth)
When $v=45ft/s$,
$D=\frac{45^{2}\sin(42)\cos(42)+45\cos(42)\sqrt ((45\sin(42))^{2}+64*7)}{32}$
$D=69.9311ft\approx69.93ft$ (rounded to the nearest hundredth)
From this data we can see that the distance $D$ increases as the initial velocity $v$ increases.
Solving $c$,
From the answers to $a$ and $b$, we see that the most effective way to increase distance $D$ is by increasing $v$. Therefore the shot-putter should increase the initial velocity to improve the performance.
