5 Marks Questions

Question 15 Marks

The volume of a right circular cone is $9856\text{ }c{{m}^{3}}$. If the diameter of the base if $28 \ cm$, find:
$i.$ Height of the cone
$ii.$ Slant height of the cone
$iii.$ surface area of the cone.

Answer

$i.$ Diameter of cone $= 28 \ cm$
$\therefore$ Radius of cone $= 14 \ cm$
Volume of cone = $9856\text{ }c{{m}^{3}}$

$\Rightarrow \frac{1}{3}\pi {{r}^{2}}h = 9856$
$\Rightarrow \frac{1}{3}\times \frac{22}{7}\times 14\times 14\times h=9856$
$\Rightarrow h=\frac{9856\times 3\times 7}{22\times 14\times 14} = 48 \ cm$
$i.$ Slant height of cone $\left( l \right)=\sqrt{{{r}^{2}}+{{h}^{2}}}$
$=\sqrt{{{\left( 14 \right)}^{2}}+{{\left( 48 \right)}^{2}}}$
$=\sqrt{196+2304}$
$=\sqrt{2500} = 50 \ cm$
$ii.$Curved surface area of cone $= \pi rl=\frac{22}{7}\times 14\times 50$=$2200\text{ }c{{m}^{2}}$

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Question 25 Marks

What length of tarpaulin $3 \ m$ wide will be required to make conical tent of height $8 \ m$ and base radius $6 \ m$? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately $20 \ cm$. $\left( \text{Use }\pi \text{ = 3}\text{.14} \right)$

Answer

Height of the conical tent $\left( h \right)$ = 8 m and Radius of the conical tent $\left( r \right)$ = 6 m
Slant height of the tent $\left( l \right)=\sqrt{{{r}^{2}}+{{h}^{2}}}$
=$\sqrt{{{\left( 6 \right)}^{2}}+{{\left( 8 \right)}^{2}}}$
=$\sqrt{36+64}$
=$\sqrt{100}$
= 10 m
Area of tarpaulin = Curved surface area of tent = $\pi rl$ $=3.14\times 6\times 10=188.4\text{ }{{m}^{2}}$
Width of tarpaulin$ = 3 m$
Let Length of tarpaulin $= L$
$\therefore $ Area of tarpaulin = $Length\times Breadth\text{ }=\text{ }L\times 3 = 3L$
Now According to question, $3 L=188.4$
$\Rightarrow L=188.4 / 3=62.8 m$
The extra length of the material required for stitching margins and cutting is $20 cm=0.2 m$.
So the total length of tarpaulin bought is $(62.8+0.2) m =63 m$

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Question 35 Marks

Mary wants to decorate her Christmas tree. She wants to place the tree on a wooden block covered with coloured paper with picture of Santa Claus on it (see figure). She must know the exact quantity of paper to buy for this purpose. If the box has length, breadth and height as $80\ cm, 40\ cm$ and $20\ cm$ respectively, then how many square sheets of paper of side $40\ cm$ would she require?

Answer

Since mary wants to paste the paper on the outer surface of the box; the quantity of paper required would be equal to the surface area of the box which is of the shape of a cuboid.
The dimensions of the box are:
Length, $\mathrm{I}=80 \mathrm{~cm}$, Breadth, $\mathrm{b}=40 \mathrm{~cm}$, Height, $\mathrm{h}=20 \mathrm{~cm}$.
The surface area of the box $=2(\mathrm{lb}+\mathrm{bh}+\mathrm{hl})$
$=2[(80 \times 40)+(40 \times 20)+(20 \times 80)]$
$=2[3200+800+1600]$
$=2 \times 5600 \mathrm{~cm}^2=11200 \mathrm{~cm}^2$
The area of each sheet of the paper $=40 \times 40=1600 \mathrm{~cm}^2$
Therefore, number of sheets required = $\frac{surface \ area \ of \ box}{area \ of \ one \ sheet \ of \ paper}$ = $\frac{11200}{1600}$ = 7

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Maths 5 Marks Questions

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