Let's solve these questions one by one.

Q1: Find the ratio of the perimeters of two squares given their area ratio is 49:169.

Given:

• The ratio of the areas of two squares is 49:169.

Let the sides of the squares be $a_1$ and $a_2$. The ratio of the areas of the squares is given by: $\frac{a_1^2}{a_2^2} = \frac{49}{169}$

Taking the square root of both sides to find the ratio of the sides: $\frac{a_1}{a_2} = \frac{\sqrt{49}}{\sqrt{169}} = \frac{7}{13}$

The ratio of the perimeters of the squares, which is proportional to their sides, is also 7:13.

Answer: $\text{a. 7:13}$

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Q2: Find the correct statement for the cuboid with dimensions 16m, 14m, and 7m.

Given:

• Length $l = 16 \, \text{m}$
• Breadth $b = 14 \, \text{m}$
• Height $h = 7 \, \text{m}$

1. Volume of the cuboid: $\text{Volume} = l \times b \times h = 16 \times 14 \times 7 = 1568 \, \text{cubic meters}$

2. Lateral Surface Area (LSA): $\text{LSA} = 2h(l + b) = 2 \times 7 \times (16 + 14) = 2 \times 7 \times 30 = 420 \, \text{square meters}$

3. Ratio of Volume and LSA: $\frac{\text{Volume}}{\text{LSA}} = \frac{1568}{420} = \frac{392}{105}$

All of these calculations are correct.

Answer: $\text{d. All of the above}$

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Q3: Find the length of the longest pole that can be placed in a room with dimensions 12m, 8m, and 9m.

The longest pole that can be placed in a room corresponds to the diagonal of the cuboid. The length of the diagonal $d$ of the cuboid is given by: $d = \sqrt{l^2 + b^2 + h^2} = \sqrt{12^2 + 8^2 + 9^2} = \sqrt{144 + 64 + 81} = \sqrt{289} = 17 \, \text{m}$

Answer: $\text{b. 17 m}$

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Q4: Find the diameter of a sphere if the ratio of its volume to surface area is 27 cm.

Given:

• Ratio of volume to surface area = 27 cm.

The volume $V$ and surface area $A$ of a sphere are given by: $V = \frac{4}{3} \pi r^3$ $A = 4 \pi r^2$ Given: $\frac{V}{A} = 27$

Substitute the formulas: $\frac{\frac{4}{3} \pi r^3}{4 \pi r^2} = 27$

Simplify: $\frac{r}{3} = 27 \quad \Rightarrow \quad r = 81 \, \text{cm}$

The diameter $D$ of the sphere is: $D = 2r = 2 \times 81 = 162 \, \text{cm}$

Answer: $\text{c. 162 cm}$

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Q5: Find the height of a cylinder with radius 7 cm and total surface area of 968 square cm.

Given:

• Radius $r = 7 \, \text{cm}$
• Total Surface Area $A = 968 \, \text{cm}^2$

The total surface area $A$ of a cylinder is given by: $A = 2\pi r (r + h)$

Substitute the given values: $968 = 2 \times 22/7 \times 7 \times (7 + h)$

Simplify: $968 = 44 \times (7 + h)$

Divide both sides by 44: $22 = 7 + h \quad \Rightarrow \quad h = 15 \, \text{cm}$

Answer: $\text{c. 15 cm}$

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Related Questions:

1. How do you find the area ratio of two squares from their side length ratio?
2. What is the formula for the lateral surface area of a cuboid?
3. How does the diagonal formula change if the cuboid's height is doubled?
4. What would the ratio of volume to surface area be if the radius of the sphere was halved?
5. How does the total surface area of a cylinder change with a different height?
6. What is the volume of the cuboid if the breadth was reduced by 50%?
7. What is the relation between radius and height for the cylinder's surface area?
8. How does the volume of the sphere compare to its surface area?

Tip: For geometric problems, visualizing the shape and writing down all relevant formulas can help in solving problems more accurately.