Respuesta :
Answer:
The required function is [tex]y=-25\cos (\pi x)+30[/tex].
The rider is 50ft above the ground at x=45.56.
Step-by-step explanation:
The general cosine function is
[tex]y=A\cos (Bx+C)+D[/tex] Â Â Â Â Â Â .... (1)
where, |A| altitude, [tex]\frac{2\pi}{B}[/tex] is period, C/B is phase shift and D is midline.
The center of the wheel is 30ft above the ground. So, D=30.
A Ferris wheel 50ft in diameter makes one revolution every 2 minutes.
[tex]\frac{2\pi}{B}=2\Rightarrow B=\pi[/tex]
Minimum value = 30-25 = 5
It means the equation passes through the point (0,5) and phase shift is 0.
[tex]5=A\cos (B(0)+(0))+30[/tex]
[tex]5-30=A\cos (0)[/tex]
[tex]-25=A[/tex]
Substitute A=-25, B=Ï€, C=0 and D=30 in equation (1).
[tex]y=-25\cos (\pi x+0)+30[/tex]
[tex]y=-25\cos (\pi x)+30[/tex]
Therefore, the required function is [tex]y=-25\cos (\pi x)+30[/tex].
Substitute y=50 to find the value of x.
[tex]50=-25\cos (\pi x)+30[/tex]
[tex]50-30=-25\cos (\pi x)[/tex]
[tex]20=-25\cos (\pi x)[/tex]
Divide both sides by -25.
[tex]\dfrac{20}{-25}=\cos (\pi x)[/tex]
[tex]-0.8=\cos (\pi x)[/tex]
[tex]\cos ^{-1}(-0.8)=\pi x[/tex]
[tex]143.13=\pi x[/tex]
Divide both sides by π.
[tex]\frac{143.13}{\pi}= x[/tex]
[tex]45.55969= x[/tex]
[tex]x\approx 45.56[/tex]
Therefore, the rider is 50ft above the ground at x=45.56.
