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The altitude of an aeroplane is 500 metres, and the angle of elevation from the runway to the aeroplane is 15^(@). Find the horizontal distance from the aeroplane to the runway, to the nearest centimetre.

The altitude of an aeroplane is $500$ metres, and the angle of elevation from the runway to the aeroplane is $15^{\circ}$ . Find the horizontal distance from the aeroplane to the runway, to the nearest centimetre.

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Find the rate of change of one variable when rate of change of other variable is given

Full solution

Q. The altitude of an aeroplane is

500

metres, and the angle of elevation from the runway to the aeroplane is

15^{\circ}

. Find the horizontal distance from the aeroplane to the runway, to the nearest centimetre.

Given Data: Given: $\newline$ Altitude of aeroplane $h = 500$ meters $\newline$ Angle of elevation $\theta = 15$ degrees $\newline$ We need to find the horizontal distance $d$ from the aeroplane to the runway. $\newline$ We will use the trigonometric relation $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ , where the opposite side is the altitude and the adjacent side is the horizontal distance we want to find.
Convert to Radians: First, we need to express the angle of elevation in radians because some calculators require angles in radians for trigonometric functions. However, since we are using the tangent function, and most modern calculators can work with degrees, we can skip this step and directly use the angle in degrees.
Apply Tangent Function: Now, we apply the tangent function to the angle of elevation to find the horizontal distance. $\newline$ $\tan(\theta) = \frac{h}{d}$ $\newline$ $\tan(15 \text{ degrees}) = \frac{500}{d}$
Solve for Horizontal Distance: We solve for $d$ by rearranging the equation: $d = \frac{h}{\tan(15^\circ)}$
Calculate Horizontal Distance: We calculate the horizontal distance using the given altitude and the tangent of $15$ degrees: $d = \frac{500}{\tan(15^\circ)}$
Convert to Centimeters: Using a calculator, we find the value of $\tan(15^\circ)$ and then divide $500$ by this value to find $d$ . $d \approx \frac{500}{0.26795}$ (value of $\tan(15^\circ)$ ) $d \approx 1865.438$ meters
Round to Nearest Centimeter: The question asks for the distance to the nearest centimeter, so we convert meters to centimeters by multiplying by $100$ . $d \approx 1865.438 \, \text{meters} \times 100 \, \text{centimeters/meter}$ $d \approx 186543.8 \, \text{centimeters}$
Round to Nearest Centimeter: The question asks for the distance to the nearest centimeter, so we convert meters to centimeters by multiplying by $100$ . $d \approx 1865.438$ meters $* 100$ centimeters/meter $d \approx 186543.8$ centimetersWe round the distance to the nearest centimeter. $d \approx 186544$ centimeters

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