Wile E. Coyote catapults himself off of a 500 ft. cliff with an initial velocity of 20 ft/sec.
His height t seconds after catapulting is given by the function h(t) = −16t2 + 20t + 500 .
(Note the y-intercept of (0, 500).
a. Find his maximum height (Hint: he reaches maximum height at the top of the parabola.)
b. Find how long it takes for Wile E. to land (unhurt, of course). (Hint: what’s his height
when he lands?)
c. What’s another (more general) name for the point on the graph where Wile E. lands?
(a) recall that the vertex occurs at t = -b/2a = 20/32 = 5/8
So, figure h(5/8)
(b) solve for t when h=0
(c) the point where he lands is a root of the function. h(t) = 0
a. To find the maximum height, we need to find the vertex of the parabola represented by the function h(t) = −16t^2 + 20t + 500.
The vertex of a parabola of the form h(t) = at^2 + bt + c is given by the coordinates (-b/2a, h(-b/2a)). In this case, a = -16 and b = 20.
So, the x-coordinate of the vertex is -20/(2*(-16)) = 20/32 = 5/8.
To find the y-coordinate, substitute this value of t into the function: h(5/8) = -16(5/8)^2 + 20(5/8) + 500 = -25 + 25/2 + 500 = -25 + 50/2 + 500 = 175.
Therefore, the maximum height reached by Wile E. Coyote is 175 feet.
b. To find how long it takes for Wile E. to land, we need to find the value of t when his height is 0 (i.e., h(t) = 0).
Solving the equation -16t^2 + 20t + 500 = 0, we can either factor or use the quadratic formula. However, in this case, the quadratic formula is simpler.
Using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values a = -16, b = 20, and c = 500, we get:
t = (-20 ± √(20^2 - 4(-16)(500))) / (2*(-16))
t = (-20 ± √(400 + 32000)) / (-32)
t = (-20 ± √32400) / (-32)
t = (-20 ± 180) / (-32)
Simplifying further, we have two possible solutions:
t₁ = (-20 + 180) / (-32) = 160 / -32 = -5
t₂ = (-20 - 180) / (-32) = -200 / -32 = 25/4.
Since time cannot be negative (as it is measured in seconds), the correct solution is t = 25/4.
Therefore, it takes Wile E. Coyote 25/4 seconds (or 6.25 seconds) to land.
c. Another name for the point on the graph where Wile E. lands is the x-intercept. The x-intercept is the value of t when the height, h(t), is equal to zero. In this case, the x-intercept represents the time when Wile E. lands.