genevieve is going to throw a rock from the top of a trail overlooking the ocean. When she throws the rock upward from 160 feet above the ocean, the function h (t) = -16t2+48t+160 models the height, h, of the rock above the ocean as a function of time, t. Find a. the zeros of this function that tell us when the rock will hit the ocean. b. when the rock will be 160 feet above the ocean. c. the height of the rock at t=1.5 seconds.

see explanation

Step-by-step explanation:

Given

h (t) = - 16t² + 48t + 160

(a)

To find the zeros equate h (t) to zero, that is

- 16t² + 48t + 160 = 0 (divide through by - 16)

t² - 3t - 10 = 0 ← in standard form

(t - 5) (t + 2) = 0 ← in factored form

Equate each factor to zero and solve for t

t - 5 = 0 ⇒ t = 5

t + 2 = 0 ⇒ t = - 2

However t > 0 ⇒ t = 5 is when the rock hits the ocean

(b)

Equate h (t) to 160

- 16t² + 48t + 160 = 160 (subtract 160 from both sides)

- 16t² + 48t = 0 (divide through by - 16)

t² - 3t = 0 ← factor out t from each term

t (t - 3) = 0

t = 0 ← time when rock was thrown from the top of the trail

t - 3 = 0 ⇒ t = 3

The rock is 160 ft above the ocean at 0 seconds and 3 seconds

(c)

Substitute t = 1.5 into h (t)

h (1.5) = - 16 (1.5) ² + 48 (1.5) + 160 = - 36 + 72 + 160 = 196

The rock is 196 ft above the ocean at 1.5 seconds

See below in bold.

Step-by-step explanation:

a. h = - 16t^2 + 48t + 160 = 0

Simplifying bt dividing through by 16:

-t^2 + 3t + 10 = 0

t^2 - 3t - 10 = 0 (standard form)

(t - 5) (t + 2) = 0

t = 5 seconds (answer). (we ignore the negative root).

b. When the rock is 160 feet above the ocean:

160 = - 16t^2 + 48t + 160

-16t^2 + 48t = 0

-t^2 + 3t = 0

t^2 - 3t = 0

t (t - 3) = 0

t = 0, 3

Time is 3 seconds (answer).

c. Height at 1.5 seconds

= - 16 (1.5) ^2 + 48 (1.5) + 160

= 196 feet. (answer).