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A right triangular prism of height 18 cm and of base sides 5 cm, 12 cm and 13 cm is transformed into another right triangular prism on a base of sides 9 cm, 12 cm and 15 cm. Find the height of the new prism and the change in the whole surface area. 

(a) 10 cm, 120 cm2 

(b) 8 cm, 132 cm2

(c) 10 cm, 132 cm2

(d) 8 cm, 120 cm2

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(c) 10 cm, 132 cm2

For the first prism, a = 5 cm, b = 12 cm, c = 13 cm 

⇒ s = \(\frac{a+b+c}{2}=\frac{5+12+13}{2}\) = 15 cm

Area of the base = \(\sqrt{s(s-a)(s-b)(s-c)}\)

= \(\sqrt{15\times10\times3\times2}\) cm2 = 30 cm2

Let V1 be the volume of the prism. Then, 

V1 = Area of the base x height 

⇒ V1 = (30 × 18) cm3 = 540 cm3 

Let S1 be the total surface area of the prism. Then 

S1 = Lateral surface area + 2 (Area of base) 

= (Perimeter of the base × height) + 2 (Area of base) 

= (30 × 18 + 2 × 30) cm2 = 600 cm2

Let h be the height of new prism. Then, for the new prism 

a = 9 cm, b = 12 cm, c = 15 cm ⇒ s = \(\frac{9+12+15}{2}=18\) cm

Area of the base = \(\sqrt{s(s-a)(s-b)(s-c)}\)

= \(\sqrt{18\times9\times6\times3}\) cm2 = 54 cm2

∴ V2 = Volume of new prism = (54 × h) cm3 

V1 = V2 ⇒ 54 × h = 540 ⇒ h = 10 cm. 

Let S2 be the total surface area of the new prism. Then, 

S2 = (36 × 10 + 2 × 54) cm2 = 468 cm2 

Change in the whole surface area = S1 – S2 

= (600 – 468) cm2 = 132 cm2.

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