# A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is opened at the top. Ignoring the thickness of the plastic sheet,

(i) Determine the area of the sheet

(ii) The cost of the sheet for it, if a sheet measuring \[1\text{ }{{\text{m}}^{2}}\] costs Rs. 20.

Last updated date: 01st Apr 2023

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Answer

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Hint: First of all, get the area of the plastic sheet required by filling the total surface area of the open box that is \[S=2\left( hb+hl \right)+lb\] and then multiply this area with the cost of sheet per \[{{\text{m}}^{2}}\] to get the total cost of sheet.

Complete step-by-step answer:

We are given a plastic box of length, breadth and height equal to 1.5 m, 1.25 m and 65 cm respectively. It is opened at the top. We have to find the area of the sheet. Also, we have to find the cost of the sheet if the sheet costs Rs. 20/\[{{\text{m}}^{2}}\].

(i) First of all, we have to find the area of the sheet required to make this box. To get the total area of the sheet required, we have to find the total surface area of the given box. Let us consider the surface area of the box to be S.

We know that this box is in the shape of a cuboid. So, the total surface area of cuboid = 2 (lb + bh + hl) where l, b and h are the length, breadth and height of the cuboid.

But as we are given that this box is an open box that is it is open at the top. So, we must subtract the area of the upper face from the total surface area of the cuboid to get the required surface area of the given box. So, we get

Surface area of box (S) = 2 (lb + bh + hl) – (Area of the upper face)

We know that area of the upper face of box = l x b

So, we get, S = 2(lb + bh + hl) – lb

Therefore, we get surface area of the box (S) = 2 (bh + hl) + lb …..(1)

Now, we know that the area of the plastic sheet required = Surface area of box (1)

So, now we substitute the length of box = 1.5 m, breadth of the box = 1.25 m and height of the box = 65 cm = 0.65 m in equation (1) to get the plastic sheet required. So, we get,

S = Area of plastic sheet required \[=2\left[ \left( 1.25\times 0.65 \right)+\left( 0.65 \right)\left( 1.5 \right) \right]+\left( 1.5\times 1.25 \right)\text{ }{{\text{m}}^{2}}\]

\[S=\left[ 2\left( 0.8125+0.975 \right)+1.875 \right]\text{ }{{\text{m}}^{2}}\]

\[S=5.45\text{ }{{\text{m}}^{2}}\]

So, we get the area of the plastic sheet required to make the given box equal to \[5.45\text{ }{{\text{m}}^{2}}\].

(ii) Now, we are given that, cost of sheet = Rs.20/\[{{\text{m}}^{2}}\]

So, to get the total cost of the sheet required, we have to multiply the area of the total sheet required with cost of the sheet per \[{{\text{m}}^{2}}\]. So, we get,

Total cost of the sheet required to make this box = Rs (20 x 5.45) = Rs.109.

So, we get the total cost of the sheet to make this as equal to Rs.109.

Note: Here, many students forget to subtract the area of the upper face of the cuboid box. So, this must be kept in mind whenever the box is open. Also, students must note that whenever we calculate the surface area, volume etc. of any object, we must substitute length, breadth and height in the same unit. In this question, we can see that the given height is in centimeters. So, we must first convert it into meters and then only substitute it to get the surface area.

Complete step-by-step answer:

We are given a plastic box of length, breadth and height equal to 1.5 m, 1.25 m and 65 cm respectively. It is opened at the top. We have to find the area of the sheet. Also, we have to find the cost of the sheet if the sheet costs Rs. 20/\[{{\text{m}}^{2}}\].

(i) First of all, we have to find the area of the sheet required to make this box. To get the total area of the sheet required, we have to find the total surface area of the given box. Let us consider the surface area of the box to be S.

We know that this box is in the shape of a cuboid. So, the total surface area of cuboid = 2 (lb + bh + hl) where l, b and h are the length, breadth and height of the cuboid.

But as we are given that this box is an open box that is it is open at the top. So, we must subtract the area of the upper face from the total surface area of the cuboid to get the required surface area of the given box. So, we get

Surface area of box (S) = 2 (lb + bh + hl) – (Area of the upper face)

We know that area of the upper face of box = l x b

So, we get, S = 2(lb + bh + hl) – lb

Therefore, we get surface area of the box (S) = 2 (bh + hl) + lb …..(1)

Now, we know that the area of the plastic sheet required = Surface area of box (1)

So, now we substitute the length of box = 1.5 m, breadth of the box = 1.25 m and height of the box = 65 cm = 0.65 m in equation (1) to get the plastic sheet required. So, we get,

S = Area of plastic sheet required \[=2\left[ \left( 1.25\times 0.65 \right)+\left( 0.65 \right)\left( 1.5 \right) \right]+\left( 1.5\times 1.25 \right)\text{ }{{\text{m}}^{2}}\]

\[S=\left[ 2\left( 0.8125+0.975 \right)+1.875 \right]\text{ }{{\text{m}}^{2}}\]

\[S=5.45\text{ }{{\text{m}}^{2}}\]

So, we get the area of the plastic sheet required to make the given box equal to \[5.45\text{ }{{\text{m}}^{2}}\].

(ii) Now, we are given that, cost of sheet = Rs.20/\[{{\text{m}}^{2}}\]

So, to get the total cost of the sheet required, we have to multiply the area of the total sheet required with cost of the sheet per \[{{\text{m}}^{2}}\]. So, we get,

Total cost of the sheet required to make this box = Rs (20 x 5.45) = Rs.109.

So, we get the total cost of the sheet to make this as equal to Rs.109.

Note: Here, many students forget to subtract the area of the upper face of the cuboid box. So, this must be kept in mind whenever the box is open. Also, students must note that whenever we calculate the surface area, volume etc. of any object, we must substitute length, breadth and height in the same unit. In this question, we can see that the given height is in centimeters. So, we must first convert it into meters and then only substitute it to get the surface area.

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