Question

$\: {\underline{\boxed{\frak{\orange{~Question~:}}}}}$

Through a rectangular field of length 90m and breadth 60m, two roads are constructed which are parallel to the sides and cut each other at right angles through the centre of the fields. If the width of each road is 3m, Find:

(i) the area covered by the roads
(ii) the cost of constructing the roads at the rate of ₹110 per m²


➠ Don't Spam
➠ Need full Explanation
➠ Attach Diagram if possible !
​

Answer 1

Given:

• Dimensions of a rectangular field: Length = 90 m and Breadth = 60 m.
• Two roads are constructed which are parallel to the sides and cut each other at right angles through the centre of the fields.
• Width of the roads is 3 m.

To find:

• The area covered by the roads.
• The cost of constructing the roads at the rate of 110/- per m².

Formulae used:

• Area of square = a² sq.units
• Area of rectangle = lb sq.units

Solution:

Look at the figure 1 in the attachment! It shows the complete visual of the field. Now, have a look on figure 2! It visualise us the diagram of intersecting roads. We're asked to find the area of roads. So, we can find it by separating the roads into parts as shown in figure 3, and summing up the area of separate parts. As second question, we're asked to find the cost of constructing roads at a given rate. How can we find it? It's really simple! We can calculate it by multiplying the area of roads with cost per sq.metre. Let's do it !!

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↪️Separation of the road into parts:

We can separate the roads into five parts as shown below!

• Two rectangles (EFGH) & (IJKL) with dimensions: l = 43.5 m and b = 3m.
• Two rectangles (MHIN) & (LGOP) with dimensions: l = 28.5 m and b = 3 m.
• A square (CDGH) with side = 3 m.

$\:$

↪️Finding the area of each part:

As rectangles EFGH and IJKL have same dimensions, their areas will also be same. So we can find the area of one rectangle and multiply it with two.

$\\ \:\:\:\:\longmapsto{\bf{2(Area\:of\:a\: rectangle)}}$

$\\ \:\:\:\:\longmapsto{\bf{2(lb)}}$

$\\ \:\:\:\:\longmapsto{\bf{2(43.5\times 3)}}$

$\\ \:\:\:\:\longmapsto{\bf{2(130.5)}}$

$\\ \:\:\:\:\longmapsto\underline{\bf{\purple{261\:m^2}}}$

$\:$

We know that the dimensions of rectangles MHIN and LGOP are same. So, we can carry out the step by following the method we've used to find the areas of rectangles EFGH and IJKL.

$\\ \:\:\:\:\longmapsto{\bf{2(lb)}}$

$\\ \:\:\:\:\longmapsto{\bf{2(28.5\times 3)}}$

$\\ \:\:\:\:\longmapsto{\bf{2(85.5)}}$

$\\ \:\:\:\:\longmapsto\underline{\bf{\blue{171\:m^2}}}$

$\:$

Now, we shall find the area of the square part, i.e, the area of HIGL! By inserting the measure of side in the formula:

$\\ \:\:\:\:\longmapsto{\bf{a^2\:sq.units}}$

$\\ \:\:\:\:\longmapsto{\bf{(3)^2}}$

$\\ \:\:\:\:\longmapsto{\bf{3\times 3}}$

$\\ \:\:\:\:\longmapsto\underline{\bf{\pink{9\:m^2}}}$

$\:$

↪️Area of the roads:

By summing up obtained measures of the separated parts of the roads:

$\\ \:\:\:\:\longmapsto{\bf{Area\:_{(EFGH+IJKL+MHIN+LGOP+HIGL)}}}$

$\\ \:\:\:\:\longmapsto{\bf{261+171+9}}$

$\\ \:\:\:\:\longmapsto{\bf{432+9}}$

$\\ \:\:\:\:\longmapsto{\boxed{\boxed{\frak{\red{441\:m^2}}}}}$

$\:$

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↪️Cost of constructing the roads:

We know that,

• The area of the roads = 441 m²
• Cost of constructing roads per sq.metre = 110/-

So, we can find the cost of constructing by multiplying 441 with 110, i.e.,

$\\ \:\:\:\:\longmapsto{\bf{441\times 110}}$

$\\ \:\:\:\:\longmapsto{\boxed{\boxed{\sf{\red{Rs.\:\frak{48,510}}}}}}$

$\\ \therefore\underline{\sf{The\:area\:of\:the\: roads\:is\:441\:m^2\:\&\:the\:cost\:of\: constructing\:them\:is\:Rs.\:48,510.}}$

Answer 2

Refer to attachment