salvadorperez26
21.03.2020 •
Mathematics
The concentration of a drug t hours after being injected is given by
C(t) = (0.3t)/(t^2+5). Find the time when the concentration is at a maximum. Give your answer accurate to at least 2 decimal places.
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Ответ:
The concentration is at maximum at t = 2.24 hours which is obtained from the first derivative of the function at 0
Given the function :
C(t) = (0.3t)/(t²+5)The maximum value of the function is given by the first derivative of C(t) = 0
The first derivative of C(t) using the quotient rule can be expressed thus : [V(du/dt) - U(dv/dt)] ÷ V²U = 0.3t ; V = t² + 5 ;dv/dt = 2t ; du/dt = 0.3Therefore,
dC/dt = [t² - 5(0.3) - 0.3t(2t)] ÷ (t² + 5)²
dC/dt = 0.3t² - 1.5 - 0.6t² / (t² + 5)²
Concentration will be maximum when, dC/dt = 0
Therefore,
0.3t² - 1.5 - 0.6t² / (t² + 5)² = 0
0.3t² - 1.5 - 0.6t² = 0
-0.3t² - 1.5 = 0
-0.3t² = 1.5
t² = (1.5 ÷ 0.3)
t² = 5
t = √5
t = 2.236 hours
Therefore, the concentration will be maximum in 2.24 hours.
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Ответ:
t = 2.236 hours
or 2 hours 14.16 minutes.
Step-by-step explanation:
C = (0.3t)/ (t^2 + 5) where C is the concentration of the drug.
Finding the derivative using the Quotient rule:
C' = (t^2 + 5) (0.3) - (0.3t)* 2t / (t^2 + 5)^2
C' = 0.3t^2 + 1.5 - 0.6t^2 / (t^2 + 5)^2
C' = -0.3t^2 + 1.5 / (t^2 + 5)^2
C is at a maximum when the derivative is zero.
Equating this to zero:
-0.3t^2 + 1.5 / (t^2 + 5)^2 = 0
-0.3t^2 + 1.5 = 0
-0.3t^2 = -1.5
t^2 = -1.5 / -0.3 = 5
t = √5
t = 2.236 hours.
Ответ:
Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers. The square root of a number, n, written below is the number that gives n when multiplied by itself.