Quantitative Aptitude
Mensuration
2,507 Questions
Mensuration Questions
C
Correct answer
Explanation
The tank volume is 3 * 4 * 5 = 60 cubic units. The cube volume is 3 * 3 * 3 = 27 cubic units. When the cube is placed in the tank, it displaces its own volume of water. Remaining volume = 60 - 27 = 33 cubic units.
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$6 : 5$
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$3 : 4$
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$4 : 3$
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$5 : 6$
D
Correct answer
Explanation
Ratio of areas = (r1/r2)^2 = 25/36. Ratio of radii = sqrt(25/36) = 5/6. Ratio of circumferences = 2*pi*r1 / 2*pi*r2 = r1/r2 = 5/6.
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$\pi r^2$
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$2\pi r$
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$2\pi r^2$
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$4\pi r$
A
Correct answer
Explanation
The standard formula for the area of a circle is pi * r^2.
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$(10\pm 0.1)$ $cm$
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$(10\pm 0.01)$ $cm$
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$(10\pm 0.001)$ $cm$
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$(10\pm 0.003)$ $cm$
C
Correct answer
Explanation
Volume V = s^3 = 1000. s = 10. Using error propagation, dV = 3 * s^2 * ds. 0.3 = 3 * 100 * ds, so ds = 0.3 / 300 = 0.001. Thus, s = 10 +/- 0.001.
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$\dfrac{5}{4}$
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$\dfrac{4}{5}$
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$\dfrac{5}{2}$
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$\dfrac{2}{5}$
B
Correct answer
Explanation
With fixed volume, h is proportional to 1/r^2. Minimizing the weighted material area, (5/4)pi r^2 + 2pi rh, gives h = 5r/4. Therefore, r/h = 4/5.
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$9$ & $9$
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$8$ & $10$
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$6$ & $12$
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$3$ & $15$
A
Correct answer
Explanation
A rectangle with fixed perimeter has greatest area when it is a square. With perimeter 36 m, each side is 36/4 = 9 m.
A
Correct answer
Explanation
Let base side be x and height be h. Volume V = x^2 * h = 32, so h = 32/x^2. Surface area S = x^2 + 4xh = x^2 + 4x(32/x^2) = x^2 + 128/x. To minimize, dS/dx = 2x - 128/x^2 = 0. 2x^3 = 128, x^3 = 64, x = 4. Min area = 4^2 + 128/4 = 16 + 32 = 48.
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$8\sqrt{3}$ cm
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$8$ cm
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$12\sqrt{3}$ cm
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$24$ cm
A
Correct answer
Explanation
For a cylinder of radius r and height h inscribed in a sphere of radius R, r^2 + (h/2)^2 = R^2. Volume V = pi * r^2 * h = pi * (R^2 - h^2/4) * h = pi * (R^2*h - h^3/4). Setting dV/dh = 0: R^2 - 3h^2/4 = 0, so h^2 = 4R^2/3, h = 2R/sqrt(3). With R=12, h = 24/sqrt(3) = 8*sqrt(3).
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$\displaystyle \frac{32}{3}\pi \ \mathrm{cc}$
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$72\pi \ \mathrm{cc}$
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$288\pi \ \mathrm{cc}$
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$\displaystyle \frac{288}{3}\pi \ \mathrm{cc}$
C
Correct answer
Explanation
For a cone of height h and base radius r inscribed in a sphere of radius R=9, the relationship is r^2 = R^2 - (h-R)^2. Substituting R=9, r^2 = 81 - (h-9)^2 = 18h - h^2. The volume V = (1/3)pi(r^2)h = (1/3)pi(18h^2 - h^3). Setting the derivative dV/dh = 0 gives 36h - 3h^2 = 0, so h=12. Then r^2 = 18(12) - 144 = 72. Volume = (1/3)pi(72)(12) = 288pi.
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$\dfrac{4\pi r^3}{3\sqrt{3}}$
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$\dfrac{4\pi r^3 }{3\sqrt{2}}$
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$\dfrac{\pi r^3}{3\sqrt{2}}$
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$\dfrac{4\pi r^3}{2\sqrt{3}}$
A
Correct answer
Explanation
As derived in a previous question, the volume of the largest cylinder inscribed in a sphere of radius r is 4*pi*r^3 / (3*sqrt(3)).
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$4:3$
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$3:4$
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$4:3 \pi$
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$3:4 \pi $
A
Correct answer
Explanation
Sphere volume V_s = (4/3)pi r^3. Cylinder volume V_c = pi r^2 h. Since r=h, V_c = pi r^3. Rates of increase: dVs/dt = 4 pi r^2 (dr/dt) and dVc/dt = 3 pi r^2 (dr/dt). The ratio is 4/3.
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$9.625cm^2$
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$6.125cm^2$
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$2.625cm^2$
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None of these
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$15\pi$
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$225\pi$
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$20\pi$
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$17\pi$
B
Correct answer
Explanation
Rearranging the equation: (x^2 - 10x + 25) + (y^2 + 4y + 4) = 196 + 25 + 4. This simplifies to (x-5)^2 + (y+2)^2 = 225. This is a circle with radius squared = 225, so radius = 15. Area = pi * r^2 = 225 * pi.
D
Correct answer
Explanation
Divide the equation by 4: x^2 + y^2 + 2x - 4y + lambda/4 = 0. The center is (-1, 2). The radius squared is r^2 = (-1)^2 + 2^2 - lambda/4 = 5 - lambda/4. Since area = 9pi, r^2 = 9. Thus, 5 - lambda/4 = 9, so -lambda/4 = 4, lambda = -16.
C
Correct answer
Explanation
Area A = pi * r^2, Circumference C = 2 * pi * r. dA/dr = 2 * pi * r, dC/dr = 2 * pi. dA/dC = (dA/dr) / (dC/dr) = (2 * pi * r) / (2 * pi) = r. Given r = 3, the rate is 3.