Two surveyors estimate the height of a nearby hill. One stands $5\,\text{m}$ away from the other on horizontal ground holding a measuring stick vertically. The other surveyor finds a ＂line of sight＂ to the top of the hill, and observes that this line passes the vertical stick at a height of $2.4\,\text{m}$ . They measure the distance from the stick to the top of the hill to be $1500\,\text{m}$ using laser equipment. Find, correct to the nearest metre, their estimate for the height of the hill.

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Q. Two surveyors estimate the height of a nearby hill. One stands

5\,\text{m}

away from the other on horizontal ground holding a measuring stick vertically. The other surveyor finds a ＂line of sight＂ to the top of the hill, and observes that this line passes the vertical stick at a height of

2.4\,\text{m}

. They measure the distance from the stick to the top of the hill to be

1500\,\text{m}

using laser equipment. Find, correct to the nearest metre, their estimate for the height of the hill.

Understand and Visualize: Understand the problem and visualize the scenario. $\newline$ We have a right-angled triangle where the vertical stick is perpendicular to the horizontal ground and the line of sight to the top of the hill forms the hypotenuse. The height at which the line of sight intersects the stick is $2.4\,\text{m}$ , and the distance from the stick to the top of the hill is $1500\,\text{m}$ . We need to find the height of the hill.
Use Similar Triangles: Use similar triangles to find the height of the hill. $\newline$ The small triangle formed by the vertical stick and the ground is similar to the larger triangle formed by the height of the hill and the ground. Therefore, the ratios of corresponding sides are equal. $\newline$ Let's denote the height of the hill as $h$ . $\newline$ The ratio of the height of the stick to the distance from the stick to the surveyor ( $5$ m) is equal to the ratio of the height of the hill to the total distance to the hill ( $1500$ m + $5$ m). $\newline$ So, $\frac{2.4}{5} = \frac{h}{1505}$ .
Solve Proportion: Solve the proportion to find the height of the hill. $\newline$ Cross-multiply to solve for $h$ : $\newline$ $2.4 \times 1505 = 5 \times h$ $\newline$ $3612 = 5h$
Isolate h: Divide both sides by $5$ to isolate $h$ . $\newline$ $\frac{3612}{5} = h$ $\newline$ $h = 722.4$
Round to Nearest Meter: Round the height to the nearest meter. $\newline$ The height of the hill, rounded to the nearest meter, is $722$ meters.