Answer:
a) The height of the building is 144 feet.
b) The brick hits the ground after 3 seconds.
c) The height of the brick at 2 seconds is 80 feet.
d) For Tiffany's profit:
She should charge $126 to maximize her profits.
Her maximum profit will be $56700.
Her profit will be $322300 if she charges $100.
Explanation:
For the brick's path:
a)
To determine the height of the building, we need to find the maximum value of h(t) since the brick is dropped from the building.
The maximum height can be found when the derivative of h(t) is equal to zero. Taking the derivative of h(t) with respect to t gives us h'(t) = -32t. Setting h'(t) = 0, we find that t = 0.
Therefore, the maximum height occurs at t = 0. Substituting t = 0 into h(t) gives us the height of the building: h(0) = -16(0)^2 + 144 = 144 feet.
b)
To determine when the brick hits the ground, we need to find the time when h(t) = 0. Setting h(t) = 0 and solving for t, we get -16t^2 + 144 = 0.
Dividing both sides by -16, we have t^2 - 9 = 0. Factoring, we get (t - 3)(t + 3) = 0.
This gives us two solutions: t = 3 and t = -3.
Since time cannot be negative in this context, we take t = 3 seconds as the time when the brick hits the ground.
c)
To find the height at 2 seconds, we substitute t = 2 into the equation h(t) = -16t^2 + 144: h(2) = -16(2)^2 + 144 = -64 + 144 = 80 feet.
For Tiffany's profit:
a) To maximize her profits, we need to find the value of x that gives the maximum value of p(x).
The maximum occurs at the vertex of the parabola represented by p(x).
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = -15 and b = 3780.
Plugging these values into the formula, we get x = -3780/(2*(-15)) = -3780/(-30) = 126.
Therefore, she should charge $126 to maximize her profits.
b)
To find the maximum profit, we substitute x = 126 into the equation
p(x) = -15x^2 + 3780x - 176700:
p(126) = -15(126)^2 + 3780(126) - 176700 = 56700.
Therefore, her maximum profit will be $56700.
c)
To find her profit if she charges $100, we substitute x = 100 into the equation p(x) = -15x^2 + 3780x - 176700:
p(100) = -15(100)^2 + 3780(100) - 176700 = 322300.
Therefore, her profit will be $322300 if she charges $100.
Thus,
a) The height of the building is 144 feet.
b) The brick hits the ground after 3 seconds.
c) The height of the brick at 2 seconds is 80 feet.
d) For Tiffany's profit:
She should charge $126 to maximize her profits.
Her maximum profit will be $56700.
Her profit will be $322300 if she charges $100.