## Quick solutions to Surface and Volume measurement questions of mensuration for SSC CGL Set 42

Know how to visualize the 3D and 2D shapes and apply formulas to solve 10 mensuration questions for SSC CGL Set 42 in 15 minutes.

The questions are on volumes of a cuboid, right circular cones, a sphere, a cube, a sphere; areas of a right prism with square base, a pyramid and circular wheel, triangles circles. Each question is of a different type.

If you have not taken the test yet, take it first at,

**SSC CGL level Question Set 42, Mensuration 4.**

### Solution to mensuration questions for SSC CGL Set 42 - Answering time was 15 mins

**Problem 1.**

A wooden box measures 20cm by 12cm by 10cm. Thickness of the wood is 1cm. Volume of the wood to make the box (in cm$^3$) is,

- 519
- 2400
- 960
- 1120

**Solution 1 - Problem analysis and execution**

The volume of the wood will be the outer volume of the cuboid box minus the inner volume of the holllow cuboid space which has dimensions 1cm less than the outer dimensions, that is, 19cm by 11cm by 9cm.

So the volume of the wood $=20\times{12}\times{10} - 19\times{11}\times{9}=2400-1881=519$ cm$^3$.

**Answer:** Option a: 519.

**Key concepts used:** **Cuboid volume concepts.**

**Problem 2.**

If a right circular cone is separated into solids of volumes $V_1$, $V_2$ and $V_3$ by two planes parallel to the base, which also trisect the altitude then $V_1 : V_2 : V_3$ is,

- 1 : 7 : 19
- 1 : 2 : 3
- 1 : 4 : 6
- 1 : 6 : 9

**Solution 2 - Problem analysis and visualization**

The figure below defines the problem.

The two planes parallel to the base and passing through the lines FQG and DPE trisect the height AR and also the triangle sides AB and AC. Thus,

Height of top cone, $AP=\frac{1}{3}AR$, and

height of second cone AFG, $AQ=\frac{2}{3}AR$, where AR is the height of the main cone.

Similarly because of the similarity of the three triangles $\triangle ARC$, $\triangle AQG$ and $\triangle APE$,

Base radius of the top cone, $PE=\frac{1}{3}RC$, and

base radius of the middle cone, $QG=\frac{2}{3}RC$, where RC is the base radius of the main cone.

We are to find ratio of the volumes of three parts created by the two sections parallel to the base.

If we assume the volumes of the three parts starting from the top as $V_1$, $V_2$ and $V_3$ and three cones of heights AP, AQ and AR as $V_{1c}$, $V_{2c}$ and $V_{3c}$,

$V_1 = V_{1c}$,

$V_2=V_{2c} - V_{1c}$, and

$V_3=V_{3c} - V_{2c}$.

**Solution 2 - Problem execution second stage**

The volume of a cone is,

$V=\frac{1}{3}{\pi}{r^2}{h}$, where $r$ is the base radius and $h$ is the height.

So the ratio of the volumes of the three cones is,

$V_{1c} : V_{2c} : V_{3c} = {PE^2}.{AP} : {QG^2}.{AQ} : {RC^2}.{AR}$

$=\left(\displaystyle\frac{1}{3}\right)^3 : \left(\displaystyle\frac{2}{3}\right)^3 : 1$

$=1 : 8 : 27$, a total of 36 portions.

Finally then the required ratio of the three parts,

$V_{1} : V_{2} : V_{3}=(V_{1c}) : (V_{2c}-V_{1c}) : (V_{3c}-V_{2c})$

$=1 : (8-1) : (27-8)$

$=1 : 7 : 19$.

**Answer:** Option a : 1 : 7 : 19.

**Key concepts used:** Volume concepts of cone -- similar triangle properties -- ratio concepts.

**Problem 3.**

The largest sphere that can be carved out of a cube of side 7cm has a volume (in cm$^3$),

- 481.34
- 179.67
- 543.72
- 718.66

**Solution 3 - Problem analysis and execution**

The core problem concept is shown in the figure below in which the largest circle is drawn inside a square of side length 7cm. This depicts the state in a plane through the cube, perpendicular to the top face of the cube and through the centre. The side length of the square and the radius of the circle will have the same relationship as the side length of the cube and the radius of the largest sphere inscribed in it.

The largest sphere that can be carved out of a cube will have then its radius as half the side length of the cube. In this case then the radius of the sphere is $\frac{7}{2}$cm.

Volume of the sphere is,

$V_{sphere}=\displaystyle\frac{4}{3}{\pi}{\left(\displaystyle\frac{7}{2}\right)}^3$

$=\displaystyle\frac{4}{3}\times{\displaystyle\frac{22}{7}}\times{\displaystyle\frac{7^3}{8}}$

$=\displaystyle\frac{11\times{49}}{3}$

$=\displaystyle\frac{539}{3}$

$=179.67$.

**Answer:** b: 179.67.

**Key concepts used:** Visualization -- volume of sphere -- * 3D inscribing concept -- efficient simplification -- delayed calculation*.

**Note:** We follow the concept of * efficient simplification* by delaying multiplication of numbers as late as possible and at the last stage canceling out the factors between the denominator and the numerator to reduce the calculation to its minimum. In this case we needed to multiply 49 by 11 following the rule of multiplication by 11 (adding unit's digit and ten's digit and pushing the unit's digit of the result between the two digits of the original number and adding the ten's digit of the result, if any, to the ten's digit of the original number). Finally division of 539 by 3 was a direct mental division approximating 0.66666.... to 0.67. Practice of

**efficient simplification***saves valuable seconds*and also

*of the result.*

**ensures accuracy****Problem 4.**

The height of a right prism with a square base is 15cm. If the total surface area of the prism is 608 sq cm its volume (in cm$^3$) is,

- 910
- 920
- 980
- 960

#### Solution 4 - Problem analysis and visualization

The following figure describes the problem.

A * right prism* is a three dimentional solid shape

*that*

**with a polygon base***at all*

**remains same***. In our case, as the base is a square, the top face is also a square and all horizontal cross-sections parallel to the base are squares. If the base were a triangle, the prism would have been a right triangular prism.*

**horizontal cross-sections****along the height perpendicular to the base**Thus volume of a right square prism is,

$V_{prism}=\text{Base}\times{\text{Height}}=15d^2$ where $d$ is the side length of the square.

Also the total area of the right prism in this case is,

$A_{prism}=2{d}^2+4\times{15}{d}=608$,

Or, $d^2 +30d -304=0$,

Or, $(d-8)(d+38)=0$.

As $d$ cannot be negative, $d=8$.

So the volume of the right prism is,

$V_{prism}=8^2\times{15}=960$ cm$^3$.

**Answer:** Option d: 960.

**Key concepts used:** Visualization -- **right square prism volume and area concepts -- factorization of a quadratic equation in a single variable -- basic algebraic concepts.**

**Problem 5.**

The base of a right pyramid is an equilateral triangle of side $10\sqrt{3}$ cm. If the total surface area of the pyramid is $270\sqrt{3}$ sq cm, its height is,

- $10\sqrt{3}$ cm
- $12$ cm
- $10$ cm
- $12\sqrt{3}$ cm

**Solution 5 - Problem analysis and visualization**

The following figure represents the problem visual.

By definition * a right pyramid* is a solid shape with $(n+1)$ faces where $n$ is the number of sides of the polygon base.

*with*

**Each vertex of the polygon base is connected to a single apex point**

**perpendicular to the base from the apex point passing through the centre of gravity of the polygon base.**In our right triangular pyramid, base $\triangle BCD$ is an equilateral triangle with side length as $10\sqrt{3}$ cm. All three slanting faces are isosceles triangles of same height $AQ=h_1$, say, and height of the pyramid $h$, say, is perpendicular to the base passing through the centroid G of the equilateral base. Each median such as DQ bisects opposite side BC and is perpendicular to it (as $\triangle BCD$ is an equilateral triangle).

The area of the equilateral base is,

$A_{base}=\displaystyle\frac{\sqrt{3}}{4}{d^2}$, where $d$ is the side length.

In this case,

$A_{base}=300\times{\displaystyle\frac{\sqrt{3}}{4}}=75\sqrt{3}.$

Total area of the three slanting faces is,

$A_{slanting}=3\times{\frac{1}{2}}\times{10\sqrt{3}}\times{h_1}$

$=15\sqrt{3}{h_1}$.

Total surface area of the pyramid is thus,

$A_{surface}=75\sqrt{3} + 15\sqrt{3}{h_1}=270\sqrt{3}$,

Or, $15h_1=195$,

Or, $h_1=13$.

#### Solution 5 - Problem execution second stage

Median langth of the base is,

$DQ=\sqrt{(10\sqrt{3})^2 - (5\sqrt{3})^2}=\sqrt{300-75}=15$

As centroid G divides the median in a 2 : 1 ratio,

$GQ=\frac{1}{3}\times{15}=5$.

Finally in right $\triangle AGQ$, height of pyramid,

$h=\sqrt{{h_1}^2 - GQ^2}=\sqrt{13^2 - 5^2}=12$ cm.

**Answer:** Option b: $12$ cm.

**Key concept used:** * Right pyramid concepts* -- Area of equilateral triangle concept -- median section ratio at centroid concept -- Pythagoras theorem.

**Problem 6.**

If a metallic cone of base radius 30cm and height 45cm is melted and recast into metallic spheres of radius 5cm find the maximum number of spheres that can be made,

- 81
- 41
- 40
- 80

**Solution 6 - Problem analysis and execution**

To get the maximum number of spheres of radius 5cm that can be made out of metal melted from the cone of base radius 30cm and height 45cm we need to divide the volume of the cone by the volume of a single sphere.

Volume of the cone,

$V_{cone}=\frac{1}{3}{\pi}\times{30^2}\times{45}$

$=900\times{15{\pi}}$

Volume of a single sphere is,

$V_{sphere}=\frac{4}{3}{\pi}\times{5^3}=\displaystyle\frac{500}{3}{\pi}$

So the required number of spheres is,

$N=\displaystyle\frac{V_{cone}}{V_{sphere}}$

$=\displaystyle\frac{900\times{15}\times{3}}{500}$

$=81$.

**Answer:** Option a : 81.

**Key concepts used:** Visualization -- volume of sphere and cone concepts -- **efficient simplification.**

**Problem 7.**

The diameter of a circular wheel is 7m. How many revolutions will it make in travelling 22km?

- 1000
- 100
- 400
- 500

** Solution 7 - Problem visualization and solution**

With each complete revolution of the wheel, the distance traversed will be equal to the perimeter of the wheel which is,

$\text{Perimeter}=2{\pi}\times{\displaystyle\frac{7}{2}}=22$ m.

So to cover a distance of 22 km or 22000 m, 1000 revolutions will be required.

**Answer: **a**: **1000.

**Key concepts used:** Circle perimeter -- cirular to linear distance mapping.

**Problem 8.**

In an equilateral $\triangle ABC$ of side 10cm, the side BC is trisected at D. The length of AD then (in cm) is,

- $3\sqrt{7}$
- $\displaystyle\frac{7\sqrt{10}}{3}$
- $\displaystyle\frac{10\sqrt{7}}{3}$
- $7\sqrt{3}$

** Solution 8 - Problem analysis and visualization**

The following figure represents the problem.

As D trisects BC of length 2 cm, $BD=DE=EC=\displaystyle\frac{10}{3}$ cm where E also is the second trisecting point on BC.

Also as median AP is the perpendicular bisector of side BC, it bisects section DE so that,

$DP=\displaystyle\frac{10}{6}=\frac{5}{3}$.

Median length of the equilateral $\triangle ABC$ is,

$AP=\sqrt{100-25}=5\sqrt{3}$.

Finally then in right $\triangle APD$,

$AD=\sqrt{DP^2 + AP^2}$

$=\sqrt{\displaystyle\frac{25}{9} + 75}$

$=\sqrt{\displaystyle\frac{700}{9}}$

$=\displaystyle\frac{10\sqrt{7}}{3}$.

**Answer:** Option c: $\displaystyle\frac{10\sqrt{7}}{3}$.

**Key concepts used:** Visualization -- equilateral triangle concepts -- Pythagoras theorem.

**Problem 9.**

The length of perpendiculars drawn from any point in the interior of an equilateral triangle to the respective sides are $p_1$, $p_2$ and $p_3$. The length of each side of the triangle is then,

- $\displaystyle\frac{2}{\sqrt{3}}(p_1 +p_2 +p_3)$
- $\displaystyle\frac{4}{\sqrt{3}}(p_1 +p_2 +p_3)$
- $\displaystyle\frac{1}{3}(p_1 +p_2 +p_3)$
- $\displaystyle\frac{1}{\sqrt{3}}(p_1 +p_2 +p_3)$

**Solution 9 - Problem analysis and visualization**

The following figure corresponds to visualization of the problem.

The perpendicular lengths $p_1$, $p_2$ and $p_3$ are from the internal poind D to sides AB, BC and CA of the equilateral $\triangle ABC$ of side length, say, $d$.

So sum of areas of the three triangles $\triangle DAB$, $\triangle DBC$ and $\triangle DCA$ is,

$A=\displaystyle\frac{1}{2}d(p_1 + p_2 + p_3)$.

Again, the area of the equilateral $\triangle ABC$ is,

$A=\displaystyle\frac{\sqrt{3}}{4}d^2=\displaystyle\frac{1}{2}d(p_1 + p_2 + p_3)$.

Or, $d=\displaystyle\frac{2}{\sqrt{3}}(p_1+p_2+p_3)$.

**Answer:** Option a: $\displaystyle\frac{2}{\sqrt{3}}(p_1+p_2+p_3)$.

**Key concepts used:** Visualization -- area of equilateral triangle -- area of triangle -- * technique of area decomposition into components*.

** Problem 10.**

If arcs of same length in two circles subtend angles $60^0$ and $75^0$ at their centre, the ratio of their radii is,

- 3 : 4
- 3 : 5
- 4 : 5
- 5 : 4

**Solution 10 - Problem analysis**

By definition, in a circle of radius $r$ the length of an arc AB subtending an angle $\theta$ at the centre is,

$Arc_{length}\text{ of AB}=r\theta$, where $\theta$ is in radians.

The following is the figure that depicts the relation.

This relation makes sense as we know, when the angle subtended by the arc is $2\pi$ the length of the arc is the perimeter of the circle which is, $2\pi{r}$.

#### Problem solving execution

Thus for same arc length in two circles of radii $r_1$ and $r_2$,

$r_1{\theta}_1=r_2{\theta}_2$,

Or, $r_1 : r_2 = \theta_2 : \theta_1 = 75^0 : 60^0 = 5 : 4$.

As the angles are in a ratio, the ratio of angles in radians will be same as ratio of angles in degrees.

**Answer:** d: 5 : 4.

**Key concepts used:** Arc length concept -- ratio concept.

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