Question

The figure shows a solid cuboid of dimensions 12 cm by 10 cm
by 7 cm, with a half-cylinder of diameter 4 cm horizontally
carved out of it. Find:
i. the volume of the solid;
ii. the total surface area of the solid



​

Answer 1

(i)The volume of the solid: $\bold{764.64\ cm^{3}}$

(ii)Total surface area of the solid: 472.64 cm

Explanation:

Given:

Length of the solid: 12 cm

Breath of the cuboid: 10 cm

Height of the cuboid: 7 cm

Diameter of the half cylinder: 4 cm

Radius of half cylinder: 2 cm

(i) To find the volume of the solid:

To find the volume of the solid we have calculate the total volume of the cuboid and total volume of the half cylinder then subtract the volume of half cylinder from the volume of the cuboid.

Step 1: Volume of cuboid: $length\times breath\times height$

Step 2: Substitute the given values        

$= 12cm\times 10cm\times 7cm$        

$=840\ cm^{3}$

Step 3: Volume of half cylinder: $\pi r^{2} h$

step 4:              

= $\pi* 2^{2} *12$

=75.36$cm^{3}$

Step 5: Total volume of solid = volume of cuboid - volume of half cylinder

$=840\ cm^{3}-75.36cm^{3} \\ \\=764.6\ cm^{3}$                            

(ii) To find the total surface area of the solid:  

To find the total surface area of the solid we have calculate the total surface area of cuboid and total surface area of half cylinder and then subtract the area of half cylinder from the area of cuboid

Step 1: TSA of cuboid: $2\times (lb+bh+lh)$

Step 2: substitute the given values  

$=2\times (12cm\times 10cm+10cm\times 7cm+12cm\times 7cm)$

$=2\times (120cm+70cm+84cm)$

Tsa of cuboid = 548 cm

Step 3: TSA of the half cylinder

Step 4:

$=2\pi rh /2$

$=\pi rh$

$=\pi \times 2cm\times 12cm$

=75.36 cm

Step 5: Total surface area of the solid = TSA of cuboid - TSA of half cylinde  = 548 cm - 75.36 cm  

= 472.64 cm

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