The height of a ball above the ground as a function of time is given by the function h(t)=−32t2+8t+3 where h is the height of th
1 answer:
Answer:
0.125 seconds.
Step-by-step explanation:
We have been given that the height of a ball above the ground as a function of time is given by the function , where h is the height of the ball in feet and t is the time in seconds.
We can see that our given equation is a downward opening parabola as its leading coefficient is negative. The maximum point will be vertex of parabola.
To find the time, when the ball would be at its maximum height, we need to find the x-coordinate of vertex.
Using formula , we will find the x-coordinate of vertex of parabola as:
Therefore, the ball will be at a maximum height after 0.125 seconds.
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Answer:
- h ≥ 0
- C = 0; concentration eventually decays to nothing
- (0, 0) is the only intercept in the domain. It means the concentration in the bloodstream is zero at the time the drug is injected.
- 1.95 hours
Step-by-step explanation:
1. a. The domain of the function in the context of this problem is h ≥ 0. The maximum value of h will correspond to the time at which the concentration is considered to be negligible. If that time is when the concentration decays to 1% of its peak value, then perhaps the suitable domain is 0 ≤ h ≤ 180.
b. The equation of the vertical asymptote of this function is where the denominator of C(h) is zero, that is ...
h^3 +8 = 0
h = ∛(-8)
h = -2 . . . . the equation of the vertical asymptote
This is not a concern for a medical professional because it is outside the domain of the function.
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2. As h gets large the value of the function approaches 2/h, which is to say the horizontal asymptote is C = 0. It means the concentration of medication in the blood eventually decays to zero.
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3. C(0) = 0 is the only intercept in the domain of the function. (h, C(h)) = (0, 0)
In the context of this problem, it means the blood concentration of medication is zero at the time of the injection.
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4. About 1.95 hours after injection, a maximum concentration of the drug occur in the bloodstream. This value is easily found using a graphing calculator.
Taking the derivative of the function gives you ...
C'(h) = -2(h^4 +5h^3 -16h -20)/(h^3 +8)^2
This has two real zeros and two complex zeros. The positive real zero is near h = 1.94527, about 1.95.
That is, the concentration in the bloodstream reaches a maximum about 1.95 hours after injection into the muscle.
