Math  /  Calculus

QuestionThe marginal cost (in dollars per square foot) of installing xx square feet of kitchen countertop is given by C(x)=x67C^{\prime}(x)=x^{\frac{6}{7}}. a) Find the cost of installing 60ft260 \mathrm{ft}^{2} of countertop. b) Find the cost of installing an extra 13ft213 \mathrm{ft}^{2} of countertop after 60ft260 \mathrm{ft}^{2} have already been installed. a) Set up the integral for the cost of installing 60ft260 \mathrm{ft}^{2} of countertop. C(60)=0dxC(60)=\int_{0}^{\square} \square \mathrm{dx}

Studdy Solution
Substitute the limits of integration:
C(73)C(60)=713(73)137713(60)137 C(73) - C(60) = \frac{7}{13} (73)^{\frac{13}{7}} - \frac{7}{13} (60)^{\frac{13}{7}}
The cost of installing 60ft2 60 \, \mathrm{ft}^2 of countertop is:
C(60)=713(60)137 C(60) = \frac{7}{13} (60)^{\frac{13}{7}}
The cost of installing an additional 13ft2 13 \, \mathrm{ft}^2 after 60ft2 60 \, \mathrm{ft}^2 is:
C(73)C(60)=713(73)137713(60)137 C(73) - C(60) = \frac{7}{13} (73)^{\frac{13}{7}} - \frac{7}{13} (60)^{\frac{13}{7}}

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