Math / Calculus

QuestionProblem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area $462 \mathrm{~cm}^{2}$ . Find the length of its base radius when its capacity is maximum $\left(\pi=\frac{22}{7}\right)$ . Answer

Studdy Solution

STEP 1

1. The vessel is a cylindrical shape without a lid.
2. The surface area of the metallic lamina used is $462 \, \text{cm}^2$ .
3. We are to find the radius of the base when the volume (capacity) is maximized.

STEP 2

1. Recall the formulas for the surface area and volume of a cylinder.
2. Express the height in terms of the radius using the surface area formula.
3. Substitute the expression for height into the volume formula.
4. Differentiate the volume with respect to the radius and find the critical points.
5. Determine the radius that maximizes the volume.

STEP 3

Recall the formulas for the surface area and volume of a cylinder:
Surface area (without lid): $A = \pi r^2 + 2\pi rh$
Volume: $V = \pi r^2 h$

STEP 4

Express the height $h$ in terms of the radius $r$ using the surface area formula:
Given $A = 462 \, \text{cm}^2$ , substitute into the surface area formula: $\pi r^2 + 2\pi rh = 462$
Solve for $h$ : $2\pi rh = 462 - \pi r^2$ $h = \frac{462 - \pi r^2}{2\pi r}$

STEP 5

Substitute the expression for $h$ into the volume formula:
$V = \pi r^2 \left(\frac{462 - \pi r^2}{2\pi r}\right)$ $V = \frac{r(462 - \pi r^2)}{2}$

STEP 6

Differentiate the volume $V$ with respect to $r$ and find the critical points:
$V = \frac{462r - \pi r^3}{2}$
Differentiate: $\frac{dV}{dr} = \frac{1}{2}(462 - 3\pi r^2)$
Set the derivative equal to zero to find critical points: $462 - 3\pi r^2 = 0$ $3\pi r^2 = 462$ $r^2 = \frac{462}{3\pi}$ $r = \sqrt{\frac{462}{3\pi}}$
Substitute $\pi = \frac{22}{7}$ : $r = \sqrt{\frac{462 \times 7}{3 \times 22}}$ $r = \sqrt{\frac{3234}{66}}$ $r = \sqrt{49}$ $r = 7 \, \text{cm}$

STEP 7

Verify that this radius maximizes the volume by checking the second derivative or analyzing the behavior of the function.
Since the problem asks for the radius when the capacity is maximum and $r = 7 \, \text{cm}$ is the only critical point, we conclude it maximizes the volume.
The length of the base radius when the capacity is maximum is:
$\boxed{7 \, \text{cm}}$

Was this helpful?

QuestionProblem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area $462 \mathrm{~cm}^{2}$ . Find the length of its base radius when its capacity is maximum $\left(\pi=\frac{22}{7}\right)$ . Answer

Studdy Solution

STEP 1

1. The vessel is a cylindrical shape without a lid.
2. The surface area of the metallic lamina used is $462 \, \text{cm}^2$ .
3. We are to find the radius of the base when the volume (capacity) is maximized.

STEP 2

STEP 3

Recall the formulas for the surface area and volume of a cylinder:
Surface area (without lid): $A = \pi r^2 + 2\pi rh$
Volume: $V = \pi r^2 h$

STEP 4

Express the height $h$ in terms of the radius $r$ using the surface area formula:
Given $A = 462 \, \text{cm}^2$ , substitute into the surface area formula: $\pi r^2 + 2\pi rh = 462$
Solve for $h$ : $2\pi rh = 462 - \pi r^2$ $h = \frac{462 - \pi r^2}{2\pi r}$

STEP 5

Substitute the expression for $h$ into the volume formula:
$V = \pi r^2 \left(\frac{462 - \pi r^2}{2\pi r}\right)$ $V = \frac{r(462 - \pi r^2)}{2}$

STEP 6

STEP 7

Get Studdy

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.

QuestionProblem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area 462 cm2462 \mathrm{~cm}^{2}462 cm2. Find the length of its base radius when its capacity is maximum (π=227)\left(\pi=\frac{22}{7}\right)(π=722​). Answer

Studdy Solution

STEP 1

1. The vessel is a cylindrical shape without a lid.2. The surface area of the metallic lamina used is 462 cm2 462 \, \text{cm}^2 462cm2.3. We are to find the radius of the base when the volume (capacity) is maximized.

STEP 2

STEP 3

Recall the formulas for the surface area and volume of a cylinder:Surface area (without lid): A=πr2+2πrh A = \pi r^2 + 2\pi rh A=πr2+2πrhVolume: V=πr2h V = \pi r^2 h V=πr2h

STEP 4

STEP 5

Substitute the expression for h h h into the volume formula:V=πr2(462−πr22πr) V = \pi r^2 \left(\frac{462 - \pi r^2}{2\pi r}\right) V=πr2(2πr462−πr2​) V=r(462−πr2)2 V = \frac{r(462 - \pi r^2)}{2} V=2r(462−πr2)​

STEP 6

STEP 7

Get Studdy

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.

QuestionProblem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area $462 \mathrm{~cm}^{2}$ . Find the length of its base radius when its capacity is maximum $\left(\pi=\frac{22}{7}\right)$ . Answer

1. The vessel is a cylindrical shape without a lid.
2. The surface area of the metallic lamina used is $462 \, \text{cm}^2$ .
3. We are to find the radius of the base when the volume (capacity) is maximized.

Recall the formulas for the surface area and volume of a cylinder:
Surface area (without lid): $A = \pi r^2 + 2\pi rh$
Volume: $V = \pi r^2 h$

Substitute the expression for $h$ into the volume formula:
$V = \pi r^2 \left(\frac{462 - \pi r^2}{2\pi r}\right)$ $V = \frac{r(462 - \pi r^2)}{2}$