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Math  /  Calculus

QuestionMonthly sales of a particular computer are expected to decline at the following rate of S′(t)S^{\prime}(t) computers per month, where tt is time in months and S(t)S(t) is the number of computers sold each month. S′(t)=−30t23−50S^{\prime}(t)=-30 t^{\frac{2}{3}}-50
The company plans to stop manufacturing this computer when monthly sales reach 800 computers. If monthly sales now (t=0)(t=0) are 2,060 computers, find S(t)\mathrm{S}(\mathrm{t}). Use a graphing calculator to approximate the solution of the equation S(t)=800S(t)=800. S(t)=S(t)= □\square

Studdy Solution

STEP 1

1. We are given the rate of change of sales, S′(t)=−30t23−50 S^{\prime}(t) = -30 t^{\frac{2}{3}} - 50 .
2. We know the initial condition S(0)=2060 S(0) = 2060 .
3. We need to find the function S(t) S(t) that describes the sales over time.
4. We will use a graphing calculator to approximate when S(t)=800 S(t) = 800 .

STEP 2

1. Integrate S′(t) S^{\prime}(t) to find S(t) S(t) .
2. Use the initial condition to find the constant of integration.
3. Write the expression for S(t) S(t) .
4. Use a graphing calculator to approximate the solution for S(t)=800 S(t) = 800 .

STEP 3

Integrate S′(t)=−30t23−50 S^{\prime}(t) = -30 t^{\frac{2}{3}} - 50 with respect to t t :
S(t)=∫(−30t23−50) dt S(t) = \int (-30 t^{\frac{2}{3}} - 50) \, dt

STEP 4

Perform the integration:
S(t)=∫−30t23 dt+∫−50 dt S(t) = \int -30 t^{\frac{2}{3}} \, dt + \int -50 \, dt
S(t)=−30∫t23 dt−50∫1 dt S(t) = -30 \int t^{\frac{2}{3}} \, dt - 50 \int 1 \, dt
S(t)=−30(t5353)−50t+C S(t) = -30 \left( \frac{t^{\frac{5}{3}}}{\frac{5}{3}} \right) - 50t + C
S(t)=−905t53−50t+C S(t) = -\frac{90}{5} t^{\frac{5}{3}} - 50t + C
S(t)=−18t53−50t+C S(t) = -18 t^{\frac{5}{3}} - 50t + C

STEP 5

Use the initial condition S(0)=2060 S(0) = 2060 to find C C :
2060=−18(0)53−50(0)+C 2060 = -18(0)^{\frac{5}{3}} - 50(0) + C
2060=C 2060 = C

STEP 6

Write the expression for S(t) S(t) :
S(t)=−18t53−50t+2060 S(t) = -18 t^{\frac{5}{3}} - 50t + 2060

STEP 7

Use a graphing calculator to approximate the solution for S(t)=800 S(t) = 800 :
Set the equation:
−18t53−50t+2060=800 -18 t^{\frac{5}{3}} - 50t + 2060 = 800
Rearrange:
−18t53−50t+1260=0 -18 t^{\frac{5}{3}} - 50t + 1260 = 0
Use a graphing calculator to find the value of t t that satisfies this equation.
The expression for S(t) S(t) is:
S(t)=−18t53−50t+2060 S(t) = -18 t^{\frac{5}{3}} - 50t + 2060
Use a graphing calculator to solve for t t when S(t)=800 S(t) = 800 .

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