Im having trouble getting the correct answer to this question. Could someone help out.
Monthly sales of a particular personal computer are expected to decline at the following rate of S'(t) computers per month, where t is time in months and S(t) is the number of computers sold each month.
S'(t)= -15t^2/3
The company plans to stop manufacturing this computer when monthly sales reach 600 computers. If monthly sales now (t=0) are 1140 computers,find S(t). How long will the company continue to manufacture this computer.
1140 - 15t^(2/3) = 600,
-15t^(2/3) = 600 - 1140 = - 540,
Divide both sides by -15:
t^(2/3) = 36,
Take log of both sides:
2/3logt = log36,
Multiply both sides by 3/2:
logt = 3/2log36,
logt = 2.3345,
t = 10^(2.3345) = 216 months.
To find S(t), you need to find the antiderivative of S'(t).
Taking the antiderivative of S'(t) = -15t^(2/3), you get:
∫ S'(t) dt = S(t) = -15 * ∫ t^(2/3) dt
To integrate t^(2/3), you can use the power rule of integration:
∫ t^n dt = (t^(n+1))/(n+1)
Applying the power rule, you get:
S(t) = -15 * (t^(2/3 + 1))/(2/3 + 1) + C
Simplifying, you get:
S(t) = -15 * (3/5) * t^(5/3) + C
Given that monthly sales now (t=0) are 1140 computers, you can substitute the values to find C:
S(0) = -15 * (3/5) * 0^(5/3) + C
1140 = 0 + C
C = 1140
Now replace C with 1140 in the equation for S(t):
S(t) = -15 * (3/5) * t^(5/3) + 1140
To find how long the company will continue to manufacture the computer, set S(t) = 600 and solve for t:
600 = -15 * (3/5) * t^(5/3) + 1140
Rearranging the equation to isolate t, you get:
-15 * (3/5) * t^(5/3) = 600 - 1140
-15 * (3/5) * t^(5/3) = -540
t^(5/3) = -540 * (-5/3) / (-15 * 3/5)
t^(5/3) = 60
Taking the fifth root of both sides, you get:
t = (60)^(3/5)
Evaluating t, you find:
t ≈ 8.324
Therefore, the company will continue to manufacture this computer for approximately 8.324 months.
To find the function S(t), we need to integrate S'(t) with respect to t.
S'(t) = -15t^(2/3)
To integrate -15t^(2/3) with respect to t, we add 1 to the exponent and divide by the new exponent.
∫(-15t^(2/3)) dt = ∫(-15t^(2/3) * t^(1/3)) dt
= -15 * ∫(t^(2/3 + 1/3)) dt
= -15 * ∫(t^(1)) dt
= -15 * (t^(1+1))/(1+1) + C
= -15 * t^(4/3)/4 + C
= -15/4 * t^(4/3) + C
Now, we can find the constant of integration, C, using the initial condition that S(0) = 1140.
S(0) = -15/4 * (0)^(4/3) + C
1140 = C
So, we have the function S(t) = -15/4 * t^(4/3) + 1140
To find how long the company will continue to manufacture this computer, we need to find the time, t, when S(t) = 600.
-15/4 * t^(4/3) + 1140 = 600
Subtract 1140 from both sides:
-15/4 * t^(4/3) = 600 - 1140
-15/4 * t^(4/3) = -540
Now, divide both sides by -15/4:
t^(4/3) = -540 / (-15/4)
t^(4/3) = 144
To solve for t, we need to take the fourth root of both sides:
t = (144)^(3/4)
Now, calculate the value of t to find how long the company will continue to manufacture this computer.