Math / Geometry

Question40. A cone-shaped paper drinking cup is to be made to hold $27 \mathrm{~cm}^{3}$ of water. Find the height and radius of the cup that will use the smallest amount of paper.

Studdy Solution

STEP 1

1. The cone has a volume of $27 \, \text{cm}^3$ .
2. We need to find the dimensions (height and radius) that minimize the surface area of the cone.
3. The surface area of the cone includes the lateral surface area but not the base.

STEP 2

1. Recall the formulas for the volume and surface area of a cone.
2. Express one variable in terms of the other using the volume formula.
3. Substitute into the surface area formula to express it as a function of one variable.
4. Find the critical points of the surface area function.
5. Determine which critical point minimizes the surface area.
6. Calculate the dimensions of the cone.

STEP 3

Recall the formulas for the volume and surface area of a cone:
- Volume: $V = \frac{1}{3} \pi r^2 h$ - Lateral Surface Area: $A = \pi r \sqrt{r^2 + h^2}$

STEP 4

Use the volume formula to express $h$ in terms of $r$ :
$27 = \frac{1}{3} \pi r^2 h$ $h = \frac{81}{\pi r^2}$

STEP 5

Substitute $h = \frac{81}{\pi r^2}$ into the surface area formula:
$A = \pi r \sqrt{r^2 + \left(\frac{81}{\pi r^2}\right)^2}$

STEP 6

Simplify the expression and find the derivative to locate critical points:
$A = \pi r \sqrt{r^2 + \frac{6561}{\pi^2 r^4}}$
Differentiate $A$ with respect to $r$ and set the derivative equal to zero to find critical points.

STEP 7

Solve for $r$ to find the critical points and determine which minimizes the surface area.

STEP 8

Calculate the corresponding height $h$ using $h = \frac{81}{\pi r^2}$ for the minimizing radius $r$ .
Once the calculations are complete, the height and radius that minimize the surface area will be determined.

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Question40. A cone-shaped paper drinking cup is to be made to hold $27 \mathrm{~cm}^{3}$ of water. Find the height and radius of the cup that will use the smallest amount of paper.

Studdy Solution

STEP 1

1. The cone has a volume of $27 \, \text{cm}^3$ .
2. We need to find the dimensions (height and radius) that minimize the surface area of the cone.
3. The surface area of the cone includes the lateral surface area but not the base.

STEP 2

STEP 3

Recall the formulas for the volume and surface area of a cone:
- Volume: $V = \frac{1}{3} \pi r^2 h$ - Lateral Surface Area: $A = \pi r \sqrt{r^2 + h^2}$

STEP 4

Use the volume formula to express $h$ in terms of $r$ :
$27 = \frac{1}{3} \pi r^2 h$ $h = \frac{81}{\pi r^2}$

STEP 5

Substitute $h = \frac{81}{\pi r^2}$ into the surface area formula:
$A = \pi r \sqrt{r^2 + \left(\frac{81}{\pi r^2}\right)^2}$

STEP 6

Simplify the expression and find the derivative to locate critical points:
$A = \pi r \sqrt{r^2 + \frac{6561}{\pi^2 r^4}}$
Differentiate $A$ with respect to $r$ and set the derivative equal to zero to find critical points.

STEP 7

Solve for $r$ to find the critical points and determine which minimizes the surface area.

STEP 8

Calculate the corresponding height $h$ using $h = \frac{81}{\pi r^2}$ for the minimizing radius $r$ .
Once the calculations are complete, the height and radius that minimize the surface area will be determined.

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Question40. A cone-shaped paper drinking cup is to be made to hold 27 cm327 \mathrm{~cm}^{3}27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.

Studdy Solution

STEP 1

1. The cone has a volume of 27 cm3 27 \, \text{cm}^3 27cm3.2. We need to find the dimensions (height and radius) that minimize the surface area of the cone.3. The surface area of the cone includes the lateral surface area but not the base.

STEP 2

STEP 3

Recall the formulas for the volume and surface area of a cone:- Volume: V=13πr2h V = \frac{1}{3} \pi r^2 h V=31​πr2h - Lateral Surface Area: A=πrr2+h2 A = \pi r \sqrt{r^2 + h^2} A=πrr2+h2​

STEP 4

Use the volume formula to express h h h in terms of r r r:27=13πr2h 27 = \frac{1}{3} \pi r^2 h 27=31​πr2h h=81πr2 h = \frac{81}{\pi r^2} h=πr281​

STEP 5

Substitute h=81πr2 h = \frac{81}{\pi r^2} h=πr281​ into the surface area formula:A=πrr2+(81πr2)2 A = \pi r \sqrt{r^2 + \left(\frac{81}{\pi r^2}\right)^2} A=πrr2+(πr281​)2​

STEP 6

Simplify the expression and find the derivative to locate critical points:A=πrr2+6561π2r4 A = \pi r \sqrt{r^2 + \frac{6561}{\pi^2 r^4}} A=πrr2+π2r46561​​Differentiate A A A with respect to r r r and set the derivative equal to zero to find critical points.

STEP 7

Solve for r r r to find the critical points and determine which minimizes the surface area.

STEP 8

Calculate the corresponding height h h h using h=81πr2 h = \frac{81}{\pi r^2} h=πr281​ for the minimizing radius r r r. Once the calculations are complete, the height and radius that minimize the surface area will be determined.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question40. A cone-shaped paper drinking cup is to be made to hold $27 \mathrm{~cm}^{3}$ of water. Find the height and radius of the cup that will use the smallest amount of paper.

1. The cone has a volume of $27 \, \text{cm}^3$ .
2. We need to find the dimensions (height and radius) that minimize the surface area of the cone.
3. The surface area of the cone includes the lateral surface area but not the base.

Recall the formulas for the volume and surface area of a cone:
- Volume: $V = \frac{1}{3} \pi r^2 h$ - Lateral Surface Area: $A = \pi r \sqrt{r^2 + h^2}$

Use the volume formula to express $h$ in terms of $r$ :
$27 = \frac{1}{3} \pi r^2 h$ $h = \frac{81}{\pi r^2}$

Substitute $h = \frac{81}{\pi r^2}$ into the surface area formula:
$A = \pi r \sqrt{r^2 + \left(\frac{81}{\pi r^2}\right)^2}$

Simplify the expression and find the derivative to locate critical points:
$A = \pi r \sqrt{r^2 + \frac{6561}{\pi^2 r^4}}$
Differentiate $A$ with respect to $r$ and set the derivative equal to zero to find critical points.

Solve for $r$ to find the critical points and determine which minimizes the surface area.

Calculate the corresponding height $h$ using $h = \frac{81}{\pi r^2}$ for the minimizing radius $r$ .
Once the calculations are complete, the height and radius that minimize the surface area will be determined.