Solve the following math problems: 1. A rectangular field is of 25 m long and 21 m wide. A 2.5 m wide strip is leveled all around it at the rate of ₹5 per square meter. Find the c... Solve the following math problems: 1. A rectangular field is of 25 m long and 21 m wide. A 2.5 m wide strip is leveled all around it at the rate of ₹5 per square meter. Find the cost of leveling the strip. 2. Two cross-roads, each of 2 meters wide, run at right angles through the center of a rectangular park of 72 m by 48 m such that each is parallel to one of the sides of the rectangle. Find the area of the remaining portion of the park. 3. A race track is in the form of a ring whose inner circumference is of 396 m and outer circumference is of 440 m. Find the width and area of the track. 4. A circular grassy plot of land of diameter 42 m has a path of wide 3.5 m running around it on the outside. Find the cost of graveling the path at ₹4 per square meter. 5. A sheet of paper measuring 30 cm by 20 cm. A strip of wide 4 cm is cut from it all around. Find the area of the remaining sheet and also the area of the cut-out strip. Read more

Steps to Solve

Here's a breakdown of how to solve each of the word problems:

Problem 1: Levelling the strip around a rectangular field

1. Calculate the outer dimensions of the field including the strip.

The strip is 2.5 m wide on all sides. So we add $2 \times 2.5 = 5$ m to both the length and the width.

Outer length = $25 + 5 = 30$ m

Outer width = $21 + 5 = 26$ m

2. Calculate the area of the outer rectangle.

Area of outer rectangle = length × width

$Area_{outer} = 30 \times 26 = 780 , m^2$

3. Calculate the area of the inner rectangle (the field).

The dimensions for this are given.

Area of inner rectangle = length × width

$Area_{inner} = 25 \times 21 = 525 , m^2$

4. Calculate the area of the strip.

Area of the strip = Area of outer rectangle - Area of inner rectangle

$Area_{strip} = 780 - 525 = 255 , m^2$

5. Calculate the cost of levelling the strip.

Cost = Area of the strip × Rate per square meter

$Cost = 255 \times 5 = 1275$

Problem 2: Cross-roads in a rectangular park

1. Calculate the area of the first road.

Length = length of the park = 72 m

Width = width of the road = 2 m

$Area_{road1} = 72 \times 2 = 144 , m^2$

2. Calculate the area of the second road.

Length = width of the park = 48 m

Width = width of the road = 2 m

$Area_{road2} = 48 \times 2 = 96 , m^2$

3. Calculate the area of the intersection of the two roads.

The intersection is a square with sides equal to the width of the roads.

Side = 2 m

$Area_{intersection} = 2 \times 2 = 4 , m^2$

4. Calculate the total area of the roads.

$Area_{roads} = Area_{road1} + Area_{road2} - Area_{intersection}$

$Area_{roads} = 144 + 96 - 4 = 236 , m^2$

5. Calculate the area of the rectangular park.

$Area_{park} = 72 \times 48 = 3456 , m^2$

6. Calculate the area of the remaining portion of the park.

$Area_{remaining} = Area_{park} - Area_{roads}$

$Area_{remaining} = 3456 - 236 = 3220 , m^2$

Problem 3: Race track in the form of a ring

1. Find the radii of the inner and outer circles.

Circumference of a circle is given by $C = 2\pi r$. So $r = C / (2\pi)$.

Inner radius $r_{inner} = 396 / (2\pi) \approx 63.025 , m$

Outer radius $r_{outer} = 440 / (2\pi) \approx 70.028 , m$

2. Calculate the width of the track.

Width = Outer radius - Inner radius

$Width = r_{outer} - r_{inner} \approx 70.028 - 63.025 = 7.003 , m$

3. Calculate the area of the track.

Area of the track = Area of outer circle - Area of inner circle

$Area = \pi r_{outer}^2 - \pi r_{inner}^2 = \pi (r_{outer}^2 - r_{inner}^2)$

$Area = \pi (70.028^2 - 63.025^2) \approx \pi (4903.92 - 3972.15) \approx \pi (931.77) \approx 2927.95 , m^2$

Problem 4: Circular grassy plot with a path

1. Calculate the radius of the grassy plot.

Diameter = 42 m, so radius $r_{inner} = 42 / 2 = 21 , m$

2. Calculate the radius of the outer circle including the path.

Path width = 3.5 m

Outer radius $r_{outer} = 21 + 3.5 = 24.5 , m$

3. Calculate the area of the path.

Area of the path = Area of outer circle - Area of inner circle

$Area = \pi r_{outer}^2 - \pi r_{inner}^2 = \pi (r_{outer}^2 - r_{inner}^2)$

$Area = \pi (24.5^2 - 21^2) = \pi (600.25 - 441) = \pi (159.25) \approx 499 , m^2$

4. Calculate the cost of gravelling the path.

Cost = Area × Rate per square meter

$Cost = 499 \times 4 = 1996$

Problem 5: Strip cut from a sheet of paper

1. Calculate the dimensions of the remaining sheet.

A strip of 4 cm is cut from all around. So, we subtract $2 \times 4 = 8$ cm from both the length and width.

Remaining length $= 30 - 8 = 22 , cm$

Remaining width $= 20 - 8 = 12 , cm$

2. Calculate the area of the remaining sheet.

$Area_{remaining} = 22 \times 12 = 264 , cm^2$

3. Calculate the area of the original sheet.

$Area_{original} = 30 \times 20 = 600 , cm^2$

4. Calculate the area of the cut-out strip.

Area of the cut-out strip = Area of original sheet - Area of remaining sheet

$Area_{strip} = 600 - 264 = 336 , cm^2$