Answer:
3.
a.4 feet
b.77 feet
c.8 seconds
Explanation:
We can start by noting that the given equation, d(p) = 1/4p^2 - 2p, represents the depth (in feet) at a given point in time as you swim underwater, where p is the time in seconds since you started swimming.
a. To find the deepest point underwater, we need to find the maximum of the function d(p). We can do this by finding the vertex of the parabola. The formula for the x-coordinate of the vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b/2a. In this case, a = 1/4 and b = -2, so the x-coordinate of the vertex is:
x = -b/2a = -(-2)/(2(1/4)) = 4
So the deepest point underwater occurs at p = 4 seconds. To find the depth at this point, we can substitute p = 4 into the equation:
d(4) = 1/4(4)^2 - 2(4) = -4 feet
Therefore, the deepest point underwater is 4 feet below the surface.
b. To find the depth when you have held your breath for 22 seconds, we need to evaluate the function d(p) at p = 22. We can round our answer to the nearest foot.
d(22) = 1/4(22)^2 - 2(22) = 77 feet
Therefore, the depth when you have held your breath for 22 seconds is approximately 77 feet below the surface.
c. To find how long you hold your breath to make the swim, we need to find the time it takes to swim from one side of the pool to the other. We can do this by finding the roots of the equation d(p) = 0, which represent the times when you reach the other side of the pool (i.e., when the depth is zero). We can use the quadratic formula to solve for p:
p = (-b ± sqrt(b^2 - 4ac))/2a
In this case, a = 1/4, b = -2, and c = 0, so the formula becomes:
p = (-(-2) ± sqrt((-2)^2 - 4(1/4)(0)))/(2(1/4)) = 8 or 0
The root p = 0 corresponds to the start of the swim, so the time it takes to make the swim is:
p = 8 - 0 = 8 seconds
Therefore, you hold your breath for 8 seconds to make the swim from one side of the pool to the other.