Cost Estimation

Question 1

If the inner length of a square based water tank is 3 m and height is 4 m, find how much water the tank may contain.

Solution:

In the square based water tank,

Inner length (l) = 3 m

Inner breadth (b) = 3 m

Area of the base of the tank (A) = $l^2 = (3)^2 = 9$ m²

Volume of the tank (V) = A × h = 9 × 4 = 36 m³

Volume of tank (V) = volume of water

So, the volume of water (V) = 36 m³

Again, we know that,

Amount of water in 1 cubic meter = 1000 l

∴ 36 m³ = 36 × 1000 l = 36,000 l

Hence, the amount of water that maybe contained in the tank is 36,000 l.

Question 2

The given figure NICE is a quadrilateral shaped land. The length between the corners N and C is 40 m. The length of perpendiculars from the corners E and I to the line NC are EG = 10 m and IS = 15 m respectively.

1. Write the formula to find the area of quadrilateral.
2. Find the area of the land.
3. A male worker can finish the work of labeling the ground in 2 days and he receives wage at the rate of Rs 1500 per day. After then, a female worker plants the dubo grass (Bermuda grass) on it and she receives wages at the rate of Rs. 70 per square meter. Find the total cost of plantation.

Solution:

In the quadrilateral NICE,

Length of the diagonal (d) = NC = 40 m

Length of perpendicular EG = $p_1$ = 10 m

Length of perpendicular IS = $p_2$ = 15 m

(a) The formula to find the area of quadrilateral (A) = $\frac{1}{2} \times d \, (p_1 + p_2)$

(b) Area of the ground (A)

$= \frac{1}{2} \times d \, (p_1 + p_2)$

$= \frac{1}{2} \times 40 , (10 + 15)$

$= 20 \times 25$

$= 500 \text{ m}^2$

(c)

Amount to be paid to the female worker for plantation = 70 × 500 = Rs. 35,000

Amount to be paid to the male worker for labeling the ground = 1500 × 2 = Rs. 3,000

Therefore, the total cost of dubo plantation = 35,000 + 3,000 = Rs 38,000

Question 3

A rectangular room has length 14 ft, breadth 12 ft. and height 10 ft. There are two square shaped windows of length 3 ft each and two doors of size 6 ft × 2 ft each.

1. Write the formula to find the area of 4 walls and ceiling.
2. What is the total cost of carpeting the floor at the rate of Rs. 300 per square meter?
3. What is the total cost of painting the 4 walls and ceiling excluding the doors and windows, if the rate of painting is Rs. 30 per square feet?
4. If the given rate increases by one third, by how much does the total cost of painting increase?

Solution:

In the rectangular room,

Length (l) = 14 ft

Breadth (b) = 12 ft

Height (h) = 10 ft

Length of the square shaped window = 3 ft. There are two doors of size 6 ft × 2 ft each.

(a) The formula to find out the area of 4 walls and ceiling (A) = $2h(l + b) + lb$

(b) Area of floor of the room = $l \times b = 14 \times 12 = 168$ ft²

We know,

Area of carpet = area of floor of the room = 168 ft²

Rate of carpet per square meter (R) = Rs. 300

Total cost of carpeting (T) = 300 × 168 = Rs. 50,400

(c) Area of 2 doors (A₁) = 2(6 × 2) = 24 ft²

Area of 2 windows (A₂) = 2 × (3)² = 18 ft²

Now, area of 4 walls and ceiling excluding windows and doors,

$A = 2h(l + b) + lb - A_1 - A_2$

$= 2 \times 10 , (14 + 12) + 14 \times 12 - 24 - 18$

$= 20 \times 26 + 168 - 42$

$= 520 + 168 - 42$

$= 646 \text{ ft}^2$

Total cost of painting on 4 walls and ceiling at the rate of Rs. 30 per square feet = 30 × 646 = Rs. 19,380

(d) When the rate of painting increases by one third, the new rate = $30 + \frac{1}{3} \times 30$ = Rs. 40

Then, the total cost = 40 × 646 = Rs. 25,840

Increase in total cost = 25,840 − 19,380 = Rs. 6,460

Hence, the total cost increases by Rs. 6,460.

Question 4

There are two pillars of height 10 ft each in the gate of a stadium. A pyramid with the same base and height 2 ft is placed on the top of each pillar. If the base of each pillar is 4 ft × 4 ft then,

1. draw two figures base on the given information.
2. find the slant height of the pyramid.
3. find the total surface area of pillars to be colored. Should we add the area of bases to find the total surface area for the purpose of painting? Give reason.
4. how much does it cost to paint the pillars together with pyramids at the rate of Rs. 95 per squared feet?

Solution:

Here, the height of pillar (h₁) = 10 ft

Height of pyramid (h₂) = 2 ft

Since, the base of pillar is square shaped, its length (a) = 4 ft

(a)

(b) We know,

Slant height of pyramid (l) = $\sqrt{(h_2)^2 + \left(\frac{a}{2}\right)^2}$

$= \sqrt{(2)^2 + \left(\frac{4}{2}\right)^2}$

$= \sqrt{4 + 4}$

$= 2.83 \text{ ft}$

(c) Lateral surface area of prism (A₁) = perimeter of base (P) × height (h₁)

$= 4a \times 10$

$= 4 \times 4 \times 10 = 160 \text{ ft}^2$

Lateral surface area of pyramid (A₂) = 2al = 2 × 4 × 2.83 = 22.64 ft²

Therefore, the total surface area of a pillar with a pyramid = A₁ + A₂

$= (160 + 22.64) = 182.64 \text{ ft}^2$

Total surface area of two pillars containing pyramids = 2 × 182.64 = 365.28 ft²

No, the base area of the pillar is not included in the total surface area, as it is lying on the ground and is not painted.

(d) Rate of painting (R) = Rs. 95 per squared ft.

Then, the total cost of painting (T) = R × A = 95 × 365.28 = Rs. 34,701.6