A right triangular prism of height 18 cm and of base sides 5 cm, 12 cm

1.	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
Answer» (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.

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

Answer»

(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.