Given:

1. The vessel consists of a hollow hemisphere surmounted by a hollow cylinder.

2. The diameter of the hemisphere ($d$) = $14$ cm.

3. The total height of the vessel ($H$) = $13$ cm.

To Find:

The inner surface area of the vessel.

Step 1: Determine the dimensions of the hemisphere and cylinder.

Since the hemisphere is surmounted by the cylinder, the radius of the cylinder is equal to the radius of the hemisphere.

Radius ($r$) = $\frac{\text{Diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm}$.

The height of the hemisphere is equal to its radius, $r = 7$ cm.

Let the height of the cylindrical part be $h$.

Total height ($H$) = Height of cylinder ($h$) + Height of hemisphere ($r$).

$13 \text{ cm} = h + 7 \text{ cm}$.

$h = 13 \text{ cm} - 7 \text{ cm} = 6 \text{ cm}$.

Step 2: Identify the formula for the inner surface area.

The inner surface area of the vessel is the sum of the Curved Surface Area (CSA) of the cylinder and the Curved Surface Area (CSA) of the hemisphere.

Formula for CSA of cylinder = $2\pi rh$.

Formula for CSA of hemisphere = $2\pi r^2$.

Total Inner Surface Area = $2\pi rh + 2\pi r^2 = 2\pi r(h + r)$.

Step 3: Calculate the surface area.

Using $\pi = \frac{22}{7}$:

Total Inner Surface Area = $2 \times \frac{22}{7} \times 7 \times (6 + 7)$

[Canceling the 7 in the numerator and denominator]

Total Inner Surface Area = $2 \times 22 \times (13)$

Total Inner Surface Area = $44 \times 13$

Calculation: $44 \times 10 = 440$; $44 \times 3 = 132$; $440 + 132 = 572$.

Final Answer: The inner surface area of the vessel is 572 cm².

Given:

Volume of each cube ($V$) = $64 \text{ cm}^3$.

Two such cubes are joined end to end to form a cuboid.

To Find:

The total surface area of the resulting cuboid.

Step 1: Determine the side length of the individual cubes.

Let the side length of each cube be $a$ cm.

The formula for the volume of a cube is given by:

$V = a^3$

Substituting the given volume:

$64 = a^3$

To find $a$, we take the cube root of both sides:

$a = \sqrt[3]{64}$

$a = 4 \text{ cm}$ [Since $4 \times 4 \times 4 = 64$]

Step 2: Determine the dimensions of the resulting cuboid.

When two cubes are joined end to end, the length of the resulting cuboid increases, while the breadth and height remain the same as the side of the cube.

Length ($l$) = $a + a = 4 + 4 = 8 \text{ cm}$

Breadth ($b$) = $a = 4 \text{ cm}$

Height ($h$) = $a = 4 \text{ cm}$

Step 3: Calculate the total surface area of the cuboid.

The formula for the total surface area ($TSA$) of a cuboid is:

$TSA = 2(lb + bh + lh)$

Substitute the values $l = 8$, $b = 4$, and $h = 4$ into the formula:

$TSA = 2((8 \times 4) + (4 \times 4) + (8 \times 4))$

$TSA = 2(32 + 16 + 32)$ [Performing multiplication inside the parentheses]

$TSA = 2(80)$ [Summing the values inside the parentheses: $32 + 16 + 32 = 80$]

$TSA = 160 \text{ cm}^2$

Final Answer: The total surface area of the resulting cuboid is 160 cm$^2$.

Given:

• Height of the solid cylinder ($h$) = $10$ cm
• Radius of the base of the cylinder ($r$) = $3.5$ cm
• The article is formed by scooping out a hemisphere from each end of the cylinder.

To Find:

The total surface area of the resulting wooden article.

Step 1: Understanding the Surface Area Components

The total surface area of the article consists of three parts:

1. The Curved Surface Area (CSA) of the cylinder.
2. The Curved Surface Area (CSA) of the top hemisphere.
3. The Curved Surface Area (CSA) of the bottom hemisphere.

Formulae to be used:

• CSA of cylinder = $2\pi rh$
• CSA of a hemisphere = $2\pi r^2$

Step 2: Setting up the Equation

Total Surface Area (TSA) = (CSA of cylinder) + (CSA of top hemisphere) + (CSA of bottom hemisphere)

$TSA = 2\pi rh + 2\pi r^2 + 2\pi r^2$

$TSA = 2\pi rh + 4\pi r^2$

$TSA = 2\pi r(h + 2r)$

Step 3: Substituting the Values

Given $r = 3.5$ cm and $h = 10$ cm.

$TSA = 2 \times \frac{22}{7} \times 3.5 \times (10 + 2 \times 3.5)$

[Since $\pi \approx \frac{22}{7}$]

Step 4: Performing the Calculation

First, simplify the radius term: $3.5 = \frac{7}{2}$

$TSA = 2 \times \frac{22}{7} \times \frac{7}{2} \times (10 + 7)$

$TSA = 2 \times \frac{22}{7} \times \frac{7}{2} \times (17)$

Cancel the common factors ($2$ and $7$):

$TSA = 22 \times 17$

$TSA = 374$

Step 5: Final Conclusion

The total surface area is calculated in square centimeters ($cm^2$).

Final Answer: 374 cm²

Given:

1. The toy consists of a cone mounted on a hemisphere.
2. The radius of the cone ($r$) = $3.5$ cm.
3. The radius of the hemisphere ($r$) = $3.5$ cm.
4. The total height of the toy ($H$) = $15.5$ cm.

To Find:

The total surface area (TSA) of the toy.

Step 1: Determine the height of the conical part ($h$)

The total height of the toy is the sum of the height of the cone ($h$) and the radius of the hemisphere ($r$).
$H = h + r$
$15.5 = h + 3.5$
$h = 15.5 - 3.5$
$h = 12$ cm

Step 2: Calculate the slant height of the cone ($l$)

The formula for the slant height of a cone is $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{(3.5)^2 + (12)^2}$
$l = \sqrt{12.25 + 144}$
$l = \sqrt{156.25}$
$l = 12.5$ cm

Step 3: Formulate the Total Surface Area (TSA) of the toy

The total surface area of the toy is the sum of the curved surface area (CSA) of the cone and the curved surface area (CSA) of the hemisphere.
$TSA = \text{CSA of cone} + \text{CSA of hemisphere}$
$TSA = \pi rl + 2\pi r^2$
$TSA = \pi r (l + 2r)$

Step 4: Perform the calculation

Substitute the values $r = 3.5$ cm, $l = 12.5$ cm, and $\pi = \frac{22}{7}$:
$TSA = \frac{22}{7} \times 3.5 \times (12.5 + 2(3.5))$
$TSA = \frac{22}{7} \times 3.5 \times (12.5 + 7)$
$TSA = \frac{22}{7} \times 3.5 \times 19.5$
Since $\frac{3.5}{7} = 0.5$:
$TSA = 22 \times 0.5 \times 19.5$
$TSA = 11 \times 19.5$
$TSA = 214.5$ cm$^2$

Final Answer: The total surface area of the toy is 214.5 cm$^2$.

Given:

• Height of the cylinder ($h$) = $2.4$ cm
• Diameter of the cylinder ($d$) = $1.4$ cm
• Radius of the cylinder ($r$) = $\frac{d}{2} = \frac{1.4}{2} = 0.7$ cm
• A conical cavity of the same height ($h = 2.4$ cm) and same radius ($r = 0.7$ cm) is hollowed out.

To Find:

The total surface area of the remaining solid, rounded to the nearest cm$^2$.

Step 1: Identify the components of the Total Surface Area (TSA)

When a conical cavity is hollowed out from a solid cylinder, the total surface area of the remaining solid consists of:

• The curved surface area of the cylinder ($2\pi rh$)
• The area of the circular base of the cylinder ($\pi r^2$)
• The curved surface area of the conical cavity ($\pi rl$)

Formula: $TSA = 2\pi rh + \pi r^2 + \pi rl$

Step 2: Calculate the slant height ($l$) of the cone

The slant height $l$ is given by the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$

$l = \sqrt{(0.7)^2 + (2.4)^2}$

$l = \sqrt{0.49 + 5.76}$

$l = \sqrt{6.25} = 2.5$ cm

Step 3: Calculate the individual areas

Using $\pi \approx \frac{22}{7}$:

Curved Surface Area of Cylinder = $2 \times \frac{22}{7} \times 0.7 \times 2.4 = 2 \times 22 \times 0.1 \times 2.4 = 10.56$ cm$^2$

Area of circular base = $\pi r^2 = \frac{22}{7} \times 0.7 \times 0.7 = 22 \times 0.1 \times 0.7 = 1.54$ cm$^2$

Curved Surface Area of Cone = $\pi rl = \frac{22}{7} \times 0.7 \times 2.5 = 22 \times 0.1 \times 2.5 = 5.5$ cm$^2$

Step 4: Sum the areas

$TSA = 10.56 + 1.54 + 5.5$

$TSA = 17.6$ cm$^2$

Step 5: Rounding to the nearest cm$^2$

Since $17.6$ is closer to $18$ than $17$, we round to $18$ cm$^2$.

Final Answer: 18 cm$^2$

Given:

A cubical block with side length $a = 7$ cm. A hemisphere is surmounted on top of this cube.

To Find:

1. The greatest diameter ($d$) the hemisphere can have.
2. The total surface area of the resulting solid.

Visual Representation:

Step 1: Determining the greatest diameter of the hemisphere.

The hemisphere is placed on the top face of the cube. For the hemisphere to be contained within the boundaries of the top face of the cube, its diameter cannot exceed the side length of the cube.

Since the side of the cube is $7$ cm, the maximum diameter $d$ is equal to the side of the cube.

$d = 7$ cm

Therefore, the radius $r$ of the hemisphere is:

$r = \frac{d}{2} = \frac{7}{2} = 3.5$ cm

Step 2: Formulating the Total Surface Area (TSA) of the solid.

The total surface area of the solid is composed of:

1. The total surface area of the cube ($6a^2$).

2. The curved surface area of the hemisphere ($2\pi r^2$).

3. Subtracting the area of the base of the hemisphere, as it is covered by the hemisphere and is not part of the external surface area ($\pi r^2$).

Formula: $TSA = (\text{Total Surface Area of Cube}) - (\text{Area of the circular base of hemisphere}) + (\text{Curved Surface Area of Hemisphere})$

$TSA = 6a^2 - \pi r^2 + 2\pi r^2$

$TSA = 6a^2 + \pi r^2$

Step 3: Calculating the values.

Substitute $a = 7$ cm and $r = 3.5$ cm (or $\frac{7}{2}$ cm) into the formula:

$TSA = 6(7)^2 + \pi (\frac{7}{2})^2$

$TSA = 6(49) + \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2}$

$TSA = 294 + \frac{11 \times 7}{2}$

$TSA = 294 + \frac{77}{2}$

$TSA = 294 + 38.5$

$TSA = 332.5$ cm$^2$

Final Answer: The greatest diameter of the hemisphere is 7 cm, and the total surface area of the solid is 332.5 cm$^2$.

Given:

• Shape of the tent: A cylinder surmounted by a cone.
• Height of the cylindrical part ($h_c$) = $2.1$ m.
• Diameter of the cylindrical part ($d$) = $4$ m.
• Slant height of the conical part ($l$) = $2.8$ m.
• Rate of canvas = $₹ 500$ per m$^2$.

To Find:

1. Total area of the canvas used for the tent.
2. Total cost of the canvas.

Step 1: Determine the radius of the base.

Since the diameter of the cylindrical part is $4$ m, the radius ($r$) is half of the diameter.

$r = \frac{d}{2} = \frac{4}{2} = 2$ m.

[Since the cone is mounted on the cylinder, the radius of the cone is also $r = 2$ m.]

Step 2: Calculate the surface area of the canvas.

The canvas covers the curved surface area of the cylinder and the curved surface area of the cone.

Formula for Curved Surface Area (CSA) of a cylinder = $2\pi rh_c$.

Formula for Curved Surface Area (CSA) of a cone = $\pi rl$.

Total Area ($A$) = $CSA_{cylinder} + CSA_{cone} = 2\pi rh_c + \pi rl = \pi r(2h_c + l)$.

Substituting the values ($r=2, h_c=2.1, l=2.8, \pi = \frac{22}{7}$):

$A = \frac{22}{7} \times 2 \times (2 \times 2.1 + 2.8)$

$A = \frac{44}{7} \times (4.2 + 2.8)$

$A = \frac{44}{7} \times 7$

$A = 44$ m$^2$.

Step 3: Calculate the total cost of the canvas.

Cost = Total Area $\times$ Rate per m$^2$.

Cost = $44 \times 500$.

Cost = $22,000$.

Final Answer: The total area of the canvas used is $44$ m$^2$ and the total cost of the canvas is $₹ 22,000$.

Given:

A cubical wooden block with edge length $l$. A hemispherical depression is cut out from one face of the cube such that the diameter of the hemisphere is equal to the edge of the cube ($d = l$).

To Find:

The total surface area of the remaining solid.

Step 1: Identify the components of the surface area.

The total surface area of the remaining solid consists of two parts:

1. The total surface area of the cube.

2. The curved surface area of the hemispherical depression (which is added to the total area).

3. The area of the circular top of the hemisphere (which must be subtracted from the cube's face because it is removed/hollowed out).

Step 2: Formulate the mathematical expressions.

Let the edge of the cube be $l$.

Total Surface Area of the cube = $6 \times (\text{edge})^2 = 6l^2$.

The diameter of the hemisphere is $l$, so the radius $r = \frac{l}{2}$.

Curved Surface Area (CSA) of the hemisphere = $2\pi r^2 = 2\pi \left(\frac{l}{2}\right)^2 = 2\pi \left(\frac{l^2}{4}\right) = \frac{\pi l^2}{2}$.

Area of the circular base of the hemisphere (to be subtracted) = $\pi r^2 = \pi \left(\frac{l}{2}\right)^2 = \frac{\pi l^2}{4}$.

Step 3: Calculate the total surface area of the remaining solid.

Total Surface Area = (Total Surface Area of Cube) - (Area of circular base) + (CSA of Hemisphere)

Total Surface Area = $6l^2 - \pi r^2 + 2\pi r^2$

Total Surface Area = $6l^2 + \pi r^2$

Substitute $r = \frac{l}{2}$ into the equation:

Total Surface Area = $6l^2 + \pi \left(\frac{l}{2}\right)^2$

Total Surface Area = $6l^2 + \frac{\pi l^2}{4}$

Step 4: Simplify the expression.

To add these terms, find a common denominator:

Total Surface Area = $\frac{24l^2}{4} + \frac{\pi l^2}{4}$

Total Surface Area = $\frac{l^2}{4} (24 + \pi)$

Final Answer: The total surface area of the remaining solid is $\frac{1}{4}l^2(\pi + 24)$ square units.

Given:

• The shape of the capsule consists of a central cylinder and two hemispheres at each end.
• Total length of the capsule ($L$) = $14$ mm.
• Diameter of the capsule ($d$) = $5$ mm.

To Find:

The total surface area of the medicine capsule.

Step 1: Determine the dimensions of the cylinder and hemispheres.

The diameter of the capsule is $5$ mm, so the radius ($r$) of the cylinder and the hemispheres is:

$r = \frac{d}{2} = \frac{5}{2} = 2.5$ mm.

The length of the cylindrical part ($h$) is obtained by subtracting the radii of the two hemispheres from the total length of the capsule:

$h = L - (r + r) = 14 - (2.5 + 2.5) = 14 - 5 = 9$ mm.

Step 2: Identify the formula for the total surface area.

The total surface area of the capsule is the sum of the Curved Surface Area (CSA) of the cylinder and the Curved Surface Areas of the two hemispheres.

Total Surface Area = (CSA of Cylinder) + 2 $\times$ (CSA of Hemisphere)

Formulae:

• CSA of Cylinder = $2\pi rh$
• CSA of Hemisphere = $2\pi r^2$

Step 3: Calculate the surface area.

Total Surface Area = $2\pi rh + 2(2\pi r^2)$

Total Surface Area = $2\pi rh + 4\pi r^2$

Factor out $2\pi r$:

Total Surface Area = $2\pi r(h + 2r)$

Substitute the values ($r = 2.5$, $h = 9$, $\pi \approx \frac{22}{7}$):

Total Surface Area = $2 \times \frac{22}{7} \times 2.5 \times (9 + 2(2.5))$

Total Surface Area = $2 \times \frac{22}{7} \times 2.5 \times (9 + 5)$

Total Surface Area = $2 \times \frac{22}{7} \times 2.5 \times 14$

Step 4: Final Arithmetic Calculation.

Total Surface Area = $2 \times 22 \times 2.5 \times \frac{14}{7}$

Total Surface Area = $44 \times 2.5 \times 2$

Total Surface Area = $44 \times 5$

Total Surface Area = $220$ mm$^2$

Final Answer: The total surface area of the medicine capsule is 220 mm$^2$.