Problems on Area - Solved Examples(Set 5)

Problems on Area - Solved Examples(Set 5)

21. The diagonal of the floor of a rectangular room is $7\frac{1}{2}$ feet. The shorter side of the room is $4\frac{1}{2}$ feet. What is the area of the room?
A. $27$ square feet	B. $22$ square feet
C. $24$ square feet	D. $20$ square feet

answer with explanation

Answer: Option A
Explanation:


Diagonal, $d=7\frac{1}{2}$ feet $=\frac{15}{2}$ feet
Breadth, $b=4\frac{1}{2}$ feet $=\frac{9}{2}$ feet

In the right-angled triangle PQR,
$l^2={\left(\frac{15}{2}\right)}^2-{\left(\frac{9}{2}\right)}^2=\frac{225}{4}-\frac{81}{4}=\frac{144}{4}$

$l=\sqrt{\frac{144}{4}}=\frac{12}{2}$ feet $=6$ feet

Area $=lb=6\times \frac{9}{2}=27{\text{feet}}^2$

22. The diagonal of a rectangle is $\sqrt{41}$ cm and its area is $20$ sq. cm. What is the perimeter of the rectangle?
A. $16$ cm	B. $10$ cm
C. $12$ cm	D. $18$ cm

answer with explanation

Answer: Option D
Explanation:

For a rectangle, $d^2=l^2+b^2$
where $l$ = length , $b$ = breadth and $d$ = diagonal of the of the rectangle

$d=\sqrt{41}\text{cm}{d}^2=l^2+b^2\Rightarrow l^2+b^2={\left(\sqrt{41}\right)}^2=41\cdots \left(1\right)$

Area $=lb=20$ cm²$\cdots \left(2\right)$

$(a+b{)}^2=a^2+2ab+b^2$

using the above formula, we have
$(l+b{)}^2=l^2+2lb+b^2=(l^2+b^2)+2lb=41+(2\times 20)=81$

$(l+b)=\sqrt{81}=9$ cm

perimeter = $2(l+b)=2\times 9=18$ cm

23. A tank is $25$ m long, $12$ m wide and $6$ m deep. What is the cost of plastering its walls and bottom at the rate of $75$ paise per sq.m?
A. Rs.$558$	B. Rs.$502$
C. Rs.$516$	D. Rs.$612$

answer with explanation

Answer: Option A
Explanation:

Consider a rectangular solid of length $l$, width $w$ and height $h$. Then,

$1.$ Total Surface area of the rectangular solid,
$\text{S}=2lw+2lh+2wh$ $=2(lw+lh+wh)$

$2.$ Volume of a rectangular solid, $V=lwh$

In this case, $l=25$ m, $w=12$ m, $h=6$ m
and all surface needs to be plastered except the top

Hence, total area to be plastered
= Total surface area - Area of the top face
$=(2lw+2lh+2wh)-lw=lw+2lh+2wh=(25\times 12)+(2\times 25\times 6)+(2\times 12\times 6)=300+300+144=744{\text{m}}^2$

Cost of plastering $=744\times 75$
$=55800$ paise $=\text{Rs.}558$

24. It is decided to construct a $2$ metre broad pathway around a rectangular plot on the inside. If the area of the plot is $96$ sq.m. and the rate of construction is $\text{Rs.}50$ per square metre, what will be the total cost of construction?
A. Rs.$3500$	B. Rs.$4200$
C. Insufficient Data	D. Rs.$4400$

answer with explanation

Answer: Option C
Explanation:

Let length and breadth of the rectangular plot be $l$ and $b$ respectively.
Area of the rectangular plot $=96$ sq.m.
$\Rightarrow lb=96$

Width of the pathway $=2$ m
Length of the remaining area in the plot $=(l-4)$
Breadth of the remaining area in the plot $=(b-4)$
Remaining area in the plot $=(l-4)(b-4)$

Area of the pathway
= Area of the rectangular plot - Remaining area in the plot
$=96-(l-4)(b-4)=96-(lb-4l-4b+16)=96-(96-4l-4b+16)=4l+4b-16$

We do not know the values of $l$ and $b$ and hence area of the pathway cannot be found out. So we cannot determine total cost of construction.

25. The area of a parallelogram is $72$ cm2 and its altitude is twice the corresponding base. What is the length of the base?
A. $6$ cm	B. $7$ cm
C. $8$ cm	D. $12$ cm

answer with explanation

Answer: Option A
Explanation:

Area of a parallelogram , A $=bh$
where $b$ is the base and $h$ is the height of the parallelogram.

Let base $=x$ cm.
Then, height $=2x$ cm(∵ altitude is twice the base)

Area $=x\times 2x=2x^2$

Area is given as $72$ cm²
$\Rightarrow 2x^2=72\Rightarrow x^2=36\Rightarrow x=6\text{cm}$