Exercise 28 Page 137

We want to determine how long it takes for riders to drop 60 feet. Let's first find when the Demon Drop reaches to the top of the tower. To do so, we need to find the vertex of the given quadratic function. h= - 16t^2 +64t-60 We need to identify a-, b-, and c-values of the quadratic function. h= - 16t^2 +64t-60 ⇕ h= - 16 t^2 + 64 t +( - 60) We see that a= - 16, b= 64, and c= - 60. We can now find the t- and h-coordinates of the vertex. Vertex: ( - b/2 a, h( - b/2 a) ) Let's find the t-coordinate of the vertex.

t=- b/2a

SubstituteII

a= - 16, b= 64

t=- 64/2( - 16)

▼

Evaluate right-hand side

MultPosNeg

a(- b)=- a * b

t=- 64/- 32

RemoveNegFracAndDenom

- a/- b= a/b

t=64/32

CalcQuot

Calculate quotient

t=2

To find the second coordinate, we need to substitute 2 for t in the function.

h= - 16t^2 +64t-60

Substitute

x= 2

h= - 16( 2)^2 +64( 2)-60

▼

Evaluate right-hand side

CalcPow

Calculate power

h= - 16(4) +64(2)-60

Multiply

Multiply

h= - 64 +128-60

AddSubTerms

Add and subtract terms

h=4

The vertex is (2,4). This means that after 2 seconds the Demon Drop will reach its maximum height, 4 feet above the t-axis. From that height, it will drop 60 feet. Let's draw a diagram to see what is happening.

Therefore, we need to find when the equation is equal to - 56 since this is where the bottom of the ride is located. To do so, we need to solve the equation below. - 56=- 16t^2 +64t-60 We will solve it by using the Quadratic Formula. We first rewrite it and then identify the values of a, b, and c. - 56=- 16t^2 +64t-60 ⇕ 0=- 16t^2 +64t-4 ⇕ - 16t^2+ 64t+( - 4)=0 We see that a= - 16, b= 64, and c= - 4. Let's substitute these values into the Quadratic Formula.

t=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

t=- 64±sqrt(( 64)^2-4( - 16)( - 4))/2( - 16)

▼

Evaluate right-hand side

CalcPow

Calculate power

t=- 64±sqrt(4096-4(- 16)(- 4))/2(- 16)

Multiply

Multiply

t=- 64±sqrt(4096-4(- 16)(- 4))/- 32

MultNegNegTwoPar

(- a)(- b)=a* b

t=- 64±sqrt(4096-4(64))/- 32

MultNegPos

(- a)b = - ab

t=- 64±sqrt(4096-256)/- 32

SubTerm

Subtract term

t=- 64±sqrt(3840)/- 32

The solutions for this equation are t= - 64 ± sqrt(3840)- 32. Let's separate them into the positive and negative cases.

t= - 64 ± sqrt(3840)- 32
t_1=- 64 + sqrt(3840)/- 32	t_2=- 64 - sqrt(3840)/- 32
t_1=64/32-16sqrt(15)/32	t_1=64/32+16sqrt(15)/32
t_1 ≈ 0.1	t_2 ≈ 3.9

The solutions to the equation are t_1=0.1 and t_2=3.9. These solutions means that the Demon Drop is at the height of - 56 feet at time 0.1 and 3.9. Hence, it takes 1.9 seconds for riders to drop 60 feet from the maximum height (2,4). ccc & Time (s) & Height (ft) & 3.9 & - 56 & 2 & 4 Difference: & 1.9 & - 60