Note that √(4 - t²) is defined only as long as 4 - t² ≥ 0, or -2 ≤ t ≤ 2. Then the real integral exists only if -2 ≤ x ≤ 2. (Otherwise we deal with complex numbers.)
If x = 2, then the integral corresponds to the area of a quarter-circle with radius 2. This means that the integral has a maximum value of 1/4 • π • 2² = π.
On the opposite end, if x = -2, then the integral has the same value, but the integral from 0 to -2 is equal to the negative integral from -2 to 0. So the minimum value is -π.
For all x in between, we observe that the integrand is continuous over the rest of its domain, so F(x) is continuous.
Then the range of F(x) is the interval [-π, π].
The roller coaster could be represented by the function for Choice D.
If you were to graph this function, you would find that the zeros (x-intercepts) of the function are at the points of -2 and 1. Those are the needed points in the problem.
Since this is a parabola that opens upward, the parabola must be below the axis (or support bar) between those values.
For 33 Hours, It is = $252.45
Now, for 1 Hour, It is 252.45 / 33 = 7.65
Now, For 36 hours, It would be: 7.65 * 36 = 275.4
In short, Your Answer would be: $275.4
Hope this helps!
