The Buckingham Fountain in Chicago shoots water from a nozzle at the base of the fountain. The height h, in feet, of the water above the ground t seconds after it leaves the nozzle is given by the function h(t) = –16t2 + 90t + 15. What is the maximum height of the water? Round to the nearest tenth.
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you know that the vertex of the parabola
y = ax^2+bx+c is at x = -b/2a.
So, your maximum height is at t = 90/32
Just find h(90/32)
Or, rewrite the equation by completing the square:
h(t) = –16t^2 + 90t + 15
= -16(t^2-90/16 t) + 15
= -16(t^2 - 90/16t + (90/32))^2 + 15 + 16(90/32)^2
= -16(t - 90/32)^2 + 2265/16
So, clearly the max height, reached at t = 90/32, is 2265/16
To find the maximum height of the water from the given function, we need to determine the vertex of the parabolic function h(t) = -16t^2 + 90t + 15. The vertex of a parabola in the form y = ax^2 + bx + c is given by the formula:
t = -b / (2a)
For our function h(t) = -16t^2 + 90t + 15, we have a = -16 and b = 90. Plugging these values into the formula, we get:
t = -90 / (2 * -16)
t = -90 / -32
t = 2.8125
So, the maximum height of the water occurs at approximately t = 2.8125 seconds. Now we need to calculate the height at this time.
h(t) = -16t^2 + 90t + 15
h(2.8125) = -16(2.8125)^2 + 90(2.8125) + 15
h(2.8125) = -16(7.890625) + 253.125 + 15
h(2.8125) = -126.250 + 253.125 + 15
h(2.8125) = 141.875
Therefore, the maximum height of the water from the Buckingham Fountain is approximately 141.9 feet.
To find the maximum height of the water, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 90t + 15. The vertex represents the highest point on the graph of the function, which corresponds to the maximum height of the water.
The vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by the formula t = -b / (2a), where t represents the x-coordinate of the vertex.
From the given function, we can see that a = -16 and b = 90. Plugging these values into the formula, we have:
t = -90 / (2 * -16)
t = -90 / -32
t ≈ 2.8125
Now that we have the value of t, we can substitute it back into the function h(t) to find the maximum height:
h(2.8125) = -16(2.8125)^2 + 90(2.8125) + 15 ≈ 122.5
Therefore, the maximum height of the water is approximately 122.5 feet.