Question

A metalworker makes an overflow pan by cutting equal squares with sides of length x from the

corners of a 30 cm by 20 cm piece of aluminium, as shown in the figure. The sides are then folded up and

the corners sealed. Drain pans aren’t a requirement for your washer, but they are an inexpensive and

simple way of protecting your home by catching small leaks and reducing the amount of water damage

from broken hoses. If your laundry room is upstairs, a drain pan is recommended to protect against leaks

that can seep into the rooms below.



i. Which of the following polynomial function V (x) gives the volume of the pan?

(a) 4x3

-60x2

+450x (b) 4x3

-100x2

+600x (c) 4x3

-65x2

+600x (d) 4x3

-60x2

+500x

ii. What is volume of the pan if the height is 6 cm?

(a) 518 cm3 (b) 746 cm3 (c) 648 cm3 (d) 864 cm3​

Answer 1

(i) Polynomial function for volume of overflow pan is $\bf V(x) =4 {x}^{3} - 100 {x}^{2} + 600x$

Option (b) is correct.

(ii) Volume of pan is 864 cm³.

Option (d) is correct.

Given:

A metalworker makes an overflow pan by cutting equal squares with sides of length x from the corners of a 30 cm by 20 cm piece of aluminium.

The sides are then folded up and the corners sealed. Drain pans aren’t a requirement for your washer, but they are an inexpensive and simple way of protecting your home by catching small leaks and reducing the amount of water damage from broken hoses.

If your laundry room is upstairs, a drain pan is recommended to protect against leaks that can seep into the rooms below.

To find:

i. Which of the following polynomial function V (x) gives the volume of the pan?

(a) 4x³-60x²+450x

(b) 4x³-100x²+600x

(c) 4x³-65x²+600x

(d) 4x³-60x²+500x

ii. What is volume of the pan if the height is 6 cm?

(a) 518 cm³ (b) 746 cm³ (c) 648 cm³ (d) 864 cm³

Solution:

Volume of cuboid = Length×breadth×height

(i)

Length of aluminium sheet after cutting corners of x cm= (30-2x) cm

Breadth of aluminium sheet after cutting corners of x cm= (20-2x) cm

Height of aluminium sheet= x cm

Volume of overflow pan: Length×breadth×height

Volume of overflow pan V(x)=

$(30 - 2x)(20 - 2x)x$

$V(x)= (600 - 60x - 40x + 4 {x}^{2} )x$

$V(x)= (600 - 100x + 4 {x}^{2} )x$

$\bf V(x) =4 {x}^{3} - 100 {x}^{2} + 600x$

Option (b) is correct.

(ii) Volume of overflow pan if height of pan is 6 cm.

Put the value of x in V(x)

$V(6) = 4 {(6)}^{3} - 100 {(6)}^{2} + 600(6)$

$V(6) = 4 \times 216 - 100 \times 36 + 3600$

$V(6) = 864 - 3600+ 3600$

$\bf V(6) = 864 \: {cm}^{3}$

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