Math Problem Statement

solve for max point

Solution

The text in the image reads:

"The approximate height of a rock in meters, thrown upwards off a cliff, is h(t)=5t2+10t+180h(t) = -5t^2 + 10t + 180 where tt is time in seconds. What is the maximum height of the rock?"

This problem requires finding the maximum height of the rock, which involves determining the vertex of the quadratic function h(t)=5t2+10t+180h(t) = -5t^2 + 10t + 180.

Steps to Solve:

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

  • a=5a = -5
  • b=10b = 10
  • c=180c = 180

For a quadratic function h(t)=at2+bt+ch(t) = at^2 + bt + c, the time tt at which the maximum height occurs can be found using the formula:

tmax=b2at_{\text{max}} = -\frac{b}{2a}

Substitute the values of aa and bb:

tmax=102(5)=1010=1 secondt_{\text{max}} = -\frac{10}{2(-5)} = \frac{10}{10} = 1 \text{ second}

Now, substitute tmax=1t_{\text{max}} = 1 second back into the height function to find the maximum height:

h(1)=5(1)2+10(1)+180h(1) = -5(1)^2 + 10(1) + 180

h(1)=5(1)+10+180h(1) = -5(1) + 10 + 180

h(1)=5+10+180=185 metersh(1) = -5 + 10 + 180 = 185 \text{ meters}

Answer:

The maximum height of the rock is 185 meters.

Would you like more details or have any further questions?

Related Questions:

  1. What is the significance of the coefficient aa being negative in the quadratic function?
  2. How can you determine the time when the rock hits the ground?
  3. What would happen to the maximum height if the initial velocity of the rock was doubled?
  4. How does the maximum height change if the rock is thrown from a lower cliff?
  5. Can you derive the general formula for the vertex of any quadratic function?

Tip:

When dealing with quadratic functions, remember that the vertex represents the maximum or minimum point. The sign of the coefficient aa determines whether the parabola opens upwards (minimum) or downwards (maximum).

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex of a Parabola

Formulas

Quadratic function formula h(t) = at^2 + bt + c
Vertex formula t_max = -b/(2a)

Theorems

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Suitable Grade Level

High School