Two cross roads, each of width 10 m, cut a right angles through the centre of a rectangular park of length 700 m and breadth 300 m and parallel to its sides. Find the area of the roads. Also find the area of the park excluding cross roads. Give the answer in hectares.

The answer is below

Step-by-step explanation:

The above problem is represented in the image below.

a) To find the area of the cross roads, we use the formula:

Area of cross roads = area of EFGH + area of IJKL - area of PQRS.

Area of rectangle = length * breadth; area of square = length * length

Rectangle EFGH = 300 m by 10 m;

Area of Rectangle EFGH = 300 m * 10 m = 3000 m²

Rectangle IJKL = 700 m by 10 m;

Area of Rectangle EFGH = 700 m * 10 m = 7000 m²

Square PQRS = 10 m by 10 m;

Area of Square PQRS = 10 m * 10 m = 100 m²

Area of cross roads = area of EFGH + area of IJKL - area of PQRS

Area of cross roads = 3000 m² + 7000 m² - 100 m² = 9900 m²

Area of cross roads = 9900 m²

1 hectare = 10000 m². Hence:

Area of cross roads = 9900 m² * (1 hectare / 10000 m²) = 0.99 hectare

Area of cross roads = 0.99 hectare

b) Area of the park excluding cross roads = Area of rectangle ABCD - Area of cross roads

Rectangle ABCD = 700 m by 300 m;

Area of Rectangle ABCD = 700 m * 300 m = 210000 m²

Therefore, substituting:

Area of the park excluding cross roads = 210000 m² - 9900 m² = 200100 m²

Area of the park excluding cross roads = 200100 m² * (1 hectare / 10000 m²)

Area of the park excluding cross roads = 20.01 hectare

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

f is the midpoint of