Question

"How are we ever going to build this bridge?" asks Omkar looking out across the raging river. "Let's start by finding the distance to the big rock on the other side." Melissa replies. Moving 100100100 meters along the river, Melissa looks back and measures the angle between Omkar and the big rock: 33^\circ33 ? 33, degree. Melissa then instructs Omkar to measure the angle between Melissa and the big rock. From his vantage point, Omkar sees an angle of 98^\circ98 ? 98, degree between Melissa and the big rock. What is the distance across the river from Omkar to the big rock? Do not round during your calculations. Round your final answer to the nearest meter.

Answer 1

The distance across the river from Omkar to the big rock is 131343149 meters

Explanation:

* Lets study the information in the problem

- Let Omkar position is point A

- Let Melissa position is point B

- Let big rock position is C on the other side of the river

* Now we have triangle ABC

- The distance between Omkar and Melissa is 100100100 meters

along the river

- The angle between Omkar and the big rock is angle BAC

∴ m∠BAC = 33°

- The angle between Melissa and the big rock is angle ABC

∴ m∠ABC = 98°

- The big rock is at angle C

* Now we can find the distance between Omkar and the big rock

by finding the length of side AC in the triangle

- By using the sine rule

∵ sin A/BC = sin B/AC = sin C/AB

∵ AB = 100100100 meters

∵ m∠ABC = 98°

- Lets find m∠C

∵ In any triangle the sum of the measures of the interior angles is 180°

∴ m∠A + m∠B + m∠C = 180°

∵ m∠A = 33° , m∠B = 98°

∴ 33° + 98° + m∠C = 180° ⇒ add

∴ 131° + m∠C = 180° ⇒ subtract 131 from both sides

∴ m∠C = 49°

- Now lets use the sine rule

∵ sin ABC/AC = sin C/AB

∴ sin 98/AC = sin 49/100100100 ⇒ by using cross multiplication

∴ AC = (sin 98 × 100100100) ÷ sin 49 = 131343148.8

∴ AC ≅ 131343149 meters

* The distance across the river from Omkar to the big rock is

131343149 meters

Answer 2

The distance across the river from Omkar to the big rock is 72 meters.

Explanation:

Using the given information draw as triangle as shown below.

According to angle sum property, the sum of interior angles of a triangle is 180°.

In triangle ABC,

`\angle A+\angle B+\angle C=180^(\circ)`

`98^(\circ)+33^(\circ)+\angle C=180^(\circ)`

`131^(\circ)+\angle C=180^(\circ)`

`\angle C=180^(\circ)-131^(\circ)=49^(\circ)`

The measure of angle C is 49°.

Sine formula:

`(a)/(\sin a)=(b)/(\sin b)=(c)/(\sin c)`

Using sine formula in triangle ABC, we get

`(AC)/(\sin B)=(AB)/(\sin C)`

`(AC)/(\sin 33^(\circ))=(100)/(\sin 49^(\circ))`

`AC=(100)/(\sin 49^(\circ))* \sin 33^(\circ)`

`AC=(100)/(0.7547)\cdot0.544639`

`AC=72.166`

`AC\approx 72`

Therefore the distance across the river from Omkar to the big rock is 72 meters.