When the suns altitude changes from 30 to 60 the length of the shadow of a tower decreases by 70m?

30. When the sun's altitude changes from 30° to 60°, the length of the shadow of a tower decreases by 70m. What is the height of the tower?
A. 60.6 m	B. 140 m
C. 35 m	D. 20.2 m

Answer: Option A

Explanation:

Let AD be the tower, BD be the initial shadow and CD be the final shadow.

Given that BC = 70 m,

ABD = 30°,

ACD = 60°,

Let CD = x, AD = h

From the right

CDA,

$\tan 60° = \dfrac{\text{AD}}{\text{CD}}\\\sqrt{3} = \dfrac{\text{h}}{\text{x}}\quad \cdots(eq:1)$

From the right

BDA,

$\tan 30° = \dfrac{\text{AD}}{\text{BD}}\\\dfrac{1}{\sqrt{3}} = \dfrac{\text{h}}{70 + \text{x}}\quad \cdots(eq:2)$$\dfrac{eq:1}{eq:2} \Rightarrow \dfrac{\sqrt{3}}{\left(\dfrac{1}{\sqrt{3}}\right)}=\dfrac{\left(\dfrac{\text{h}}{\text{x}}\right)}{\left(\dfrac{\text{h}}{70+\text{x}}\right)}\\\Rightarrow 3 = \dfrac{70 + \text{x}}{\text{x}}\\\Rightarrow 2x = 70 \\\\\Rightarrow x = 35$Substituting this value of x in eq:1, we have

$\sqrt{3} = \dfrac{\text{h}}{35}\\\Rightarrow \text{h} = 35\sqrt{3} = 35 × 1.73 \\= 60.55 \approx 60.6$

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When the sun s altitude changes from 30° to 60°, the length of the shadow of a tower decreases by 70m. What is the height of the tower?

Explanation:

Let AD be the tower, BD be the initial shadow and CD be the final shadow.Given that BC = 70 m, $\angle$ABD = 30°, $\angle$ACD = 60°,Let CD = x, AD = hFrom the right $\triangle$ CDA, tan60°=$\dfrac{AD}{CD}$√3=$\dfrac{h}{x} $ ⋯(eq:1)From the right $\triangle$ BDA, tan30°=$\dfrac{AD}{BD}$$\dfrac{1}{\sqrt{3}}=\dfrac{h}{70+x} $ ⋯(eq:2)$\dfrac{eq:1}{eq:2}⇒\dfrac{\sqrt{3}}{\left( \dfrac{1}{\sqrt{3}} \right)}=\dfrac{\left( \dfrac{h}{x}\right)}{\left( \dfrac{h}{70+x}\right)}$⇒3=$\dfrac{70+x}{x}$⇒2x=70⇒x=35Substituting this value of x in eq:1, we have√3=$\dfrac{h}{35}$⇒h=35√3=35×1.73=60.55≈60.6

Next Question

A person, standing exactly midway between two towers, observes the top of the two towers at angle of elevation of 22.5° and 67.5°. What is the ratio of the height of the taller tower to the height of the shorter tower? (Given that tan 22.5° = √2−1)

When the sun's altitude changes from 30∘ to 60∘, the length of the shadow of a tower decreases by 70 m. What is the height of the tower?

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