Question

The dimensions of a rectangular box are in the ratio 2 : 3:4 and the difference between
the cost of covering it at the rate of rupee 4 and rupee4.50 per m² is * 416. Find the dimensions of
the box​

Answer 1

Answer:

Length of the rectangular box = 8 m.

Breadth of the rectangular box = 12 m.

Height of the rectangular box = 16 m.

Step-by-step explanation:

Given:

• Dimensions of the rectangular box are in the ratio 2 : 3 : 4
• The difference between the cost of covering it at the rate of Rs 4 and Rs.4.50 per m² is Rs 416

To Find:

The dimensions of the box.

Solution:

Here we have to find the dimensions of the rectangular box.

Let the length of the rectangular box be 2x.

Let the breadth of the rectangular box be 3x.

Let the height of the rectangular box be 4x.

Now let us find the total surface area of the box.

The total surface area of a cuboid id given by,

TSA of a cuboid = 2(lb + bh + lh)

where l is the length

          b is the breadth

          h is the height

Substitute the data,

TSA of cuboid = 2 (2x × 3x + 3x × 4x + 4x × 2x)

TSA of cuboid = 2 (6x² + 12x² + 8x²)

TSA of cuboid = 2 × 26x²

TSA of cuboid = 52x²

Case 1:

If the cost of covering is Rs 4 per m²

Total cost = TSA × Rate per m²

Total cost = 52x² × 4

Total cost = 208x²------(1)

Case 2:

If the cost of covering is Rs.4.50 per m²

Total cost = 52x² × 4.50

Total cost = 234x²------(2)

By given:

Equation 2 - equation 1 = 416

234x² - 208x² = 416

26x² = 416

x² = 416/26

x² = 16

x = 4

Now finding the dimensions,

Length = 2x = 2 × 4

Length of the rectangular box = 8 m.

Breadth = 3x = 3 × 4

Breadth of the rectangular box = 12 m.

Height = 4x = 4 × 4

Height of the rectangular box = 16 m.

Answer 2

Dimensions of rectangular box are in the ratio 2:3:4.

Let the -

• Length of rectangular box = 2x
• Breadth of rectangular box = 3x
• Height of rectangular box = 4x

Curved surface area of cuboid = 2(lb + bh + hl)

Substitute the known values in above formula

$\sf➣ 2[(2x × 3x) + (3x × 4x) + (4x × 2x)]</p><p>$

$\sf➣ 8 (6x² + 12x² + 8x²)$

$\sf➣ 8(26x²)$

$\sf➣ 52x² m²$

Now,

Cost of painting at Rs. 4 = 52x² × 4

$\sf \pink ↬ \red{ Rs. 208x²}$

Cost of painting at Rs. 4.50 = 52x² × 4.50

$\huge \sf \red ↬ \green{Rs. 234x²}$

Difference between costs = Rs. 416

↬ Rs. 234x² - Rs. 208x² = Rs. 416

↬ Rs. 26x² = Rs. 416

$\sf \blue↬ \red{ 26x² = 416}$

$\sf \green ↬ \red{x² = 16}$

x = 4

So,

Length of cardboard = 2x

✯ 2(4)

✯ 8 m

Breadth of cardboard = 3x

➠ 3(4)

➠ 12 m

Height of cardboard = 4x

➠ 4(4)

➠ 16 m

•°• Dimensions are 8m, 12m and 16m