Math Problem Statement

Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function h(t) = -16t^2 + 16t + 480, where t is the time in seconds and h is the height in feet. (a) How long did it take for Jason to reach his maximum height? (b) What was the highest point that Jason reached?

Solution

Let’s solve the problem step by step. The function modeling Jason's height is given as:

h(t)=16t2+16t+480h(t) = -16t^2 + 16t + 480

where tt is time in seconds, and hh is height in feet.

Part (a): Time to reach the maximum height

The given function is a quadratic equation of the form h(t)=at2+bt+ch(t) = at^2 + bt + c, where:

  • a=16a = -16
  • b=16b = 16
  • c=480c = 480

The time to reach the maximum height of a parabola is calculated using the formula:

t=b2at = -\frac{b}{2a}

Substituting a=16a = -16 and b=16b = 16:

t=162(16)=1632=0.5seconds.t = -\frac{16}{2(-16)} = -\frac{16}{-32} = 0.5 \, \text{seconds.}

Thus, it takes 0.5 seconds for Jason to reach his maximum height.


Part (b): Maximum height reached

To find the maximum height, substitute t=0.5t = 0.5 into the function h(t)h(t):

h(0.5)=16(0.5)2+16(0.5)+480h(0.5) = -16(0.5)^2 + 16(0.5) + 480

Simplify step-by-step:

  1. (0.5)2=0.25(0.5)^2 = 0.25
  2. 16(0.25)=4-16(0.25) = -4
  3. 16(0.5)=816(0.5) = 8
  4. Combine terms: h(0.5)=4+8+480=484h(0.5) = -4 + 8 + 480 = 484

Thus, the highest point Jason reached was 484 feet.


Final Answers:

(a) It took Jason 0.5 seconds to reach his maximum height.
(b) The maximum height Jason reached was 484 feet.

Would you like further details or explanations?


Follow-up Questions:

  1. How would the graph of h(t)h(t) look, and what does the vertex represent?
  2. What is the significance of the negative coefficient of t2t^2 in the equation?
  3. How would you calculate the total time Jason was in the air before hitting the water?
  4. What would happen to the maximum height if the initial constant (480) were increased?
  5. Could this quadratic function represent any motion other than Jason’s jump? Why or why not?

Tip:

The vertex form of a quadratic equation is useful for quickly identifying maximum or minimum values. Rewrite h(t)h(t) in vertex form to confirm your calculations!

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex of a Parabola

Formulas

t = -b / (2a)
h(t) = -16t^2 + 16t + 480

Theorems

Vertex Theorem of Quadratic Functions

Suitable Grade Level

Grades 9-12