6. Jason tried cliff-diving in St. Thomas while vacationing with his family. His height as a function of time could be modeled by the
function h(t) = -16t2 + 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?
for this one i got (-1/2,476)
c) Jason hit the water after how many seconds?
a) the max is on the axis of symmetry ... t = -b / 2a = -16 / (2 * -16)
b) plug result from a) into the equation to find h
c) he hits the water when h=0 ... 0 = -16t2 + 16t + 480
... use quadratic formula to find t
To find the answers to the given questions, we need to analyze the function h(t) = -16t^2 + 16t + 480. Let's break down the questions one by one.
a) To find the time it took for Jason to reach his maximum height, we need to find the vertex of the quadratic function. The vertex formula for a quadratic function in the form ax^2 + bx + c is given by x = -b/(2a).
In our case, the quadratic function is h(t) = -16t^2 + 16t + 480. Comparing this to the vertex formula, we can see that a = -16 and b = 16. Plugging these values into the formula, we get:
t = -16/(2 * -16) = -16/(-32) = 0.5 seconds
So, it took Jason 0.5 seconds to reach his maximum height.
b) To find the highest point that Jason reached, we need to evaluate the function at the time t = 0.5 seconds.
h(0.5) = -16(0.5)^2 + 16(0.5) + 480
= -16(0.25) + 8 + 480
= -4 + 8 + 480
= 476 feet
Therefore, the highest point that Jason reached was 476 feet.
c) To find the time it took for Jason to hit the water, we need to set the height function h(t) equal to zero and solve for t.
-16t^2 + 16t + 480 = 0
This is a quadratic equation, which we can solve by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
t = (-16 ± sqrt(16^2 - 4 * -16 * 480)) / (2 * -16)
= (-16 ± sqrt(256 + 30720)) / (-32)
= (-16 ± sqrt(30976)) / (-32)
= (-16 ± 176) / (-32)
From here, we have two possible solutions for t:
1) t = (-16 + 176) / (-32) = 160 / -32 = -5 seconds
2) t = (-16 - 176) / (-32) = -192 / -32 = 6 seconds
Since time cannot be negative in this context, we discard the negative solution. Therefore, Jason hit the water after 6 seconds.
In summary:
a) Jason took 0.5 seconds to reach his maximum height.
b) Jason reached a height of 476 feet.
c) Jason hit the water after 6 seconds.