The number of ice cream cones sold on a summer day can be modeled as

when the cones are sold for dollars, .
Calculator Checkpoint:
a. How many ice cream cones are sold when they are sold at 75 cents? (Round your answer to the nearest integer.)
b. How quickly is the sale of ice cream cones changing when they are sold at 75 cents? (Round your answer to three decimal places.)
cones per dollar

Ask by Chandler Hammond. in the United States

Mar 31,2025

Question

The number of ice cream cones sold on a summer day can be modeled as

when the cones are sold for dollars, .
Calculator Checkpoint:
a. How many ice cream cones are sold when they are sold at 75 cents? (Round your answer to the nearest integer.)
b. How quickly is the sale of ice cream cones changing when they are sold at 75 cents? (Round your answer to three decimal places.)
cones per dollar

Ask by Chandler Hammond. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

To find the number of ice cream cones sold when they are sold at 75 cents, we need to substitute $$ p = 0.75 $$ into the given function $$ C(p) $$ and calculate the result.

Given function:
$$ C(p) = 599.8(0.9112^{9p+14}) $$

Substitute $$ p = 0.75 $$ into the function:
$$ C(0.75) = 599.8(0.9112^{9(0.75)+14}) $$

Now, we can calculate the number of ice cream cones sold when they are sold at 75 cents.

Next, to find how quickly the sale of ice cream cones is changing when they are sold at 75 cents, we need to find the derivative of the function $$ C(p) $$ with respect to $$ p $$ and then substitute $$ p = 0.75 $$ into the derivative.

Derivative of $$ C(p) $$ with respect to $$ p $$:
$$ C'(p) = 599.8 \cdot 9 \cdot 0.9112^{9p+14} \cdot \ln(0.9112) $$

Substitute $$ p = 0.75 $$ into the derivative:
$$ C'(0.75) = 599.8 \cdot 9 \cdot 0.9112^{9(0.75)+14} \cdot \ln(0.9112) $$

Now, we can calculate the rate of change of the sale of ice cream cones when they are sold at 75 cents.

Let's calculate the values.
Calculate the value by following steps:
- step0: Calculate:
  $$599.8\times 0.9112^{9\times 0.75+14}$$
- step1: Multiply the numbers:
  $$599.8\times 0.9112^{6.75+14}$$
- step2: Add the numbers:
  $$599.8\times 0.9112^{20.75}$$
- step3: Convert the expressions:
  $$599.8\left(\frac{1139}{1250}\right)^{\frac{83}{4}}$$
- step4: Convert the expressions:
  $$\frac{2999}{5}\left(\frac{1139}{1250}\right)^{\frac{83}{4}}$$
- step5: Multiply the numbers:
  $$\frac{2999\times 1139^{\frac{83}{4}}}{5\times 1250^{\frac{83}{4}}}$$
- step6: Simplify:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}}}{5\times 1250^{\frac{83}{4}}}$$
- step7: Simplify:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}}}{5\times 1250^{20}\times 125\sqrt[4]{2^{3}}}$$
- step8: Rewrite the expression:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}}}{625\times 1250^{20}\sqrt[4]{8}}$$
- step9: Multiply by the Conjugate:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}}\times \sqrt[4]{8^{3}}}{625\times 1250^{20}\sqrt[4]{8}\times \sqrt[4]{8^{3}}}$$
- step10: Simplify:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}}\times 2^{2}\sqrt[4]{2}}{625\times 1250^{20}\sqrt[4]{8}\times \sqrt[4]{8^{3}}}$$
- step11: Multiply the numbers:
  $$\frac{11996\times 1139^{20}\sqrt[4]{1139^{3}\times 2}}{625\times 1250^{20}\sqrt[4]{8}\times \sqrt[4]{8^{3}}}$$
- step12: Multiply the numbers:
  $$\frac{11996\times 1139^{20}\sqrt[4]{1139^{3}\times 2}}{5000\times 1250^{20}}$$
- step13: Reduce the fraction:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}\times 2}}{1250\times 1250^{20}}$$
- step14: Calculate:
  $$\frac{2999\times 1139^{20}\sqrt[4]{1139^{3}\times 2}}{1250^{21}}$$
The number of ice cream cones sold when they are sold at 75 cents is approximately 87 cones.

Next, let's calculate the rate of change of the sale of ice cream cones when they are sold at 75 cents.
The rate of change of the sale of ice cream cones when they are sold at 75 cents is approximately -72.892545 cones per dollar.

Therefore, when ice cream cones are sold at 75 cents, approximately 87 cones are sold, and the rate of change of the sale is approximately -72.892545 cones per dollar.
When ice cream cones are sold at 75 cents, approximately 87 cones are sold. The rate at which the sale of ice cream cones is changing at this price is approximately -72.893 cones per dollar.

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