A diver on a platform 40 feet in height jumps upward with an initial velocity of 5ft/s. His height in h feet after t seconds is given by the function h=-16t^2+5t+50.
a. What is his maximum height? I got 50.39
b. How long will it take him to reach the surface of the water?
h=-16t^2+5t+50
50.39 = -16t^2+5t+50
-16t^2+5t+50 = 50.39
I am going to try completing the square.
(-16t^2/16)+(5/16)t = 0.39/16
t^2 + (5/16)t = .02
I'm stuck.
Is the platform really 40 feet in height, not 50 feet...? Anyway,
a. Yes, that value is right.
b.
Time it takes from platform to maximum height (I used derivatives):
dh/dt = -32t + 5
0 = -32t + 5
t = 0.15625 s
Time it takes from maximum height to surface of water:
h = vo*t - (1/2)*g*t^2
But vo = 0 because it's freefall, and g = 32.2 ft/s^2:
50.39 = 0 - (0.5)*(-32.2)*(t^2)
50.39 = 16.1(t^2)
t^2 = 3.1298
t = 1.769 s
Total time: 1.769 + 0.15625 = 1.925 s
hope this helps~ `u`
To find the time it will take for the diver to reach the surface of the water, we need to solve the equation -16t^2 + 5t + 50 = 0. Let's continue completing the square.
Step 1: Rewrite the equation and isolate the terms involving t squared and t:
-16t^2 + 5t + 50 = 0
-16t^2 + 5t = -50
Step 2: Divide the entire equation by -16 to make the coefficient of t^2 equal to 1:
t^2 - (5/16)t = 50/16
Step 3: Take half of the coefficient of t (-5/16) and square it to get (5/32)^2 = 25/1024. Add this value to both sides of the equation:
t^2 - (5/16)t + 25/1024 = 50/16 + 25/1024
Step 4: Simplify both sides of the equation:
(t - 5/32)^2 = 3200/1024 + 25/1024
(t - 5/32)^2 = 3225/1024
Step 5: Take the square root of both sides of the equation:
t - 5/32 = ±√(3225/1024)
Step 6: Simplify the square root on the right side of the equation:
t - 5/32 = ±(√3225)/(√1024)
t - 5/32 = ±57/32
Step 7: Solve for t by adding 5/32 to both sides of the equation:
t = 5/32 ± 57/32
Step 8: Simplify:
t = 5/32 + 57/32 or t = 5/32 - 57/32
Step 9: Calculate the two values of t:
t = 62/32 or t = -52/32
Step 10: Simplify:
t = 31/16 or t = -13/8
The diver will reach the surface of the water after approximately 1.94 seconds (t = 31/16) or -1.63 seconds (t = -13/8). However, since time cannot be negative in this context, we can ignore the negative solution. Therefore, the diver will take approximately 1.94 seconds to reach the surface of the water.