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Geometry
The area of a square with side length x is equal to the area of a triangle with base x. What is the altitude of the triangle?
The area of the square is x². The area of the triangle is (1/2) × base × altitude = (1/2) × x × altitude. Setting these equal: x² = (1/2) × x × altitude, solving for altitude gives altitude = 2x.
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Areas
A rectangle has a length of 15 cm and an area of 159 cm². Its area is increased to 1 1/3 times the original area by increasing only its length. What is its new perimeter?
The original area is 159 cm² and the length is 15 cm, so the width is 159 ÷ 15 = 10.6 cm. The new area is increased to 1 1/3 times the original area, which is (4/3) × 159 = 212 cm². Since only the length is increased, the new length is 212 ÷ 10.6 ≈ 20 cm. The new perimeter is 2 × (length + width) = 2 × (20 + 10.6) = 61.2 cm, which is closest to option B (60 cm). Therefore, the correct answer is B, not C.
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Geometry
The base of a right triangle is 8 and the hypotenuse is 10. Its area is _______?
Using the Pythagorean theorem, the height of the right triangle is calculated as √(10² - 8²) = √(100 - 64) = √36 = 6. The area of the right triangle is (1/2) × base × height = (1/2) × 8 × 6 = 24. Therefore, the correct answer is option D (24).
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Geometry
One side of a rectangular field is 4 m and its diagonal length is 5 m. What is the area of the field?
Given one side of the rectangle is 4 m and the diagonal is 5 m, we can use the Pythagorean theorem to find the other side: let the other side be x, then 4² + x² = 5², so 16 + x² = 25, which gives x² = 9 and x = 3 m. The area is length × width = 4 × 3 = 12 sq m.
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Area And Perimeter
A rectangular grass field is 75 m by 55 m. It has a path 2.5 m wide all around it on the outside. Find the area of the path and the cost of constructing it at Rs.2 per sq m.
The total dimensions including the path are (75 + 2*2.5) = 80 m and (55 + 2*2.5) = 60 m. The total area including the path is 80 * 60 = 4800 sq m. The area of the field alone is 75 * 55 = 4125 sq m. Therefore, the area of the path is 4800 - 4125 = 675 sq m. At Rs.2 per sq m, the cost is 675 * 2 = Rs.1350.
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Mensuration
An agricultural field is in the form of a rectangle with length 20 m and width 14 m. A pit 6 m long, 3 m wide, and 2.5 m deep is dug in a corner of the field, and the earth taken out of the pit is spread uniformly over the remaining area of the field. By how much has the level of the field been raised?
The volume of earth dug out is 6 × 3 × 2.5 = 45 m³. The total area of the field is 20 × 14 = 280 m². The pit occupies an area of 6 × 3 = 18 m², so the remaining area is 280 - 18 = 262 m². The earth is spread uniformly over this remaining area, so the increase in level = volume / area = 45 / 262 ≈ 0.1717 m = 17.17 cm. The closest option to this value is C (17.18 cm).
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Geometry
A man walked 20 m to cross a rectangular field diagonally. If the length of the field is 16 m, find the breadth of the field.
The problem involves a rectangular field where the diagonal length is 20 m and the length is 16 m. Using the Pythagorean theorem for a right triangle formed by the length, breadth, and diagonal, we have: breadth = √(diagonal² - length²) = √(20² - 16²) = √(400 - 256) = √144 = 12 m. Therefore, the breadth of the field is 12 m, which corresponds to option B.
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Mensuration
A hall 20 m long and 15 m broad is surrounded by a verandah of uniform width 2.5 m. The cost of flooring the verandah at the rate of Rs 3.50 per sq. meter is ________?
The area of the verandah is the difference between the area of the hall plus verandah and the hall alone. The total dimensions including the verandah are (20 + 2*2.5) m by (15 + 2*2.5) m = 25 m by 20 m, so area = 500 m². The hall area is 20 m × 15 m = 300 m². The verandah area is 500 - 300 = 200 m². At Rs 3.50 per m², the cost is 200 × 3.50 = Rs 700.
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Circle Area
If the diameter of a circle is increased by 100%, its area is increased by: ________?
When the diameter of a circle is increased by 100%, it doubles. Since the area of a circle is proportional to the square of the diameter, the area increases by (2)^2 = 4 times, which is a 400% increase.
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Geometry
The radii of two circular fields are in the ratio 3:5. What percent less is the area of the first field compared to the area of the second?
The area of a circle is proportional to the square of its radius. Given the radii ratio of 3:5, the ratio of their areas is 3²:5² = 9:25. The difference in area is 25 - 9 = 16 parts. To find what percent less the area of the first field is compared to the second, calculate (16/25) × 100% = 64%. Therefore, the first field's area is 64% less than the second field's area. This matches option D.