Question

Find the surface area of the bulb.

Answer 1

Step 1: Understanding the Concept:
The bulb is spherical in shape. We need to find its total surface area using its diameter.
Step 2: Key Formula or Approach:
Surface area of a sphere:
\[ \text{SA} = 4\pi r^{2} \text{ or } \pi d^{2} \]
Step 3: Detailed Explanation:
Given: Diameter \(d = 7\) cm.
Radius \(r = \frac{d}{2} = 3.5\) cm.
Using \(\pi = \frac{22}{7}\):
\[ \text{SA} = 4 \times \frac{22}{7} \times 3.5 \times 3.5 \]
\[ \text{SA} = 4 \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \]
\[ \text{SA} = \frac{4 \times 22 \times 49}{7 \times 4} = 22 \times 7 = 154 \text{ cm}^{2} \]
Step 4: Final Answer:
The surface area of the bulb is 154 \(\text{cm}^{2}\).

Answer 2

Step 1: Understanding the Concept:
To fit inside the cuboidal space with a specific gap on all sides, the diameter must be less than the dimensions of the cuboid minus the required gaps.
Step 2: Detailed Explanation:
The cuboid dimensions are: Length \(L = 24\) cm, Width \(W = 12\) cm, Height \(H = 17\) cm.
A gap of 1 cm must be left from each side. This means for each dimension, we lose \(1 + 1 = 2\) cm of available space.
1. Along the width: Max diameter \(\le 12 - 2 = 10\) cm.
2. Along the length: Max diameter \(\le 24 - 2 = 22\) cm.
3. Along the height: The bulb can be anywhere, but it's restricted by the narrower sides of the box.
The most restrictive dimension is the width (12 cm).
Thus, the maximum diameter possible is \(12 - 2 = 10\) cm.
Step 3: Final Answer:
The maximum diameter of the bulb is 10 cm.

Answer 3

Step 1: Understanding the Concept:
The lamp is cuboidal but open at the top and bottom. The area of fabric used is the lateral surface area of the cuboid. The folds mean the actual height of fabric cut is more than the height of the lamp.
Step 2: Detailed Explanation:
Dimensions of cuboid: \(24\) cm \(\times 12\) cm \(\times 17\) cm.
Perimeter of the base \(= 2 \times (24 + 12) = 2 \times 36 = 72\) cm.
The height of the lamp is 17 cm.
Since there is a fold of 2 cm on the top edge and a fold of 2 cm on the bottom edge, the total height of the fabric required is:
\[ \text{Fabric Height} = 17 + 2 + 2 = 21 \text{ cm} \]
Area of fabric \(= \text{Perimeter} \times \text{Fabric Height} \)
\[ \text{Area} = 72 \times 21 = 1512 \text{ cm}^{2} \]
Step 3: Final Answer:
The area of the fabric used is 1512 \(\text{cm}^{2}\).

Answer 4

Step 1: Understanding the Concept:
The space available inside the lamp refers to the volume enclosed by the cuboidal fabric frame.
Step 2: Key Formula or Approach:
Volume of a cuboid:
\[ V = L \times W \times H \]
Step 3: Detailed Explanation:
The dimensions of the cuboidal frame are:
Length \(L = 24\) cm
Width \(W = 12\) cm
Height \(H = 17\) cm
\[ \text{Volume} = 24 \times 12 \times 17 \]
\[ \text{Volume} = 288 \times 17 = 4896 \text{ cm}^{3} \]
Step 4: Final Answer:
The space available inside the lamp is 4896 \(\text{cm}^{3}\).