Question

1. an airplane over the pacific sights an atoll at an angle of depression of 5°. the plane is 615 meters above the water. how many kilometers is the plane from the atoll? 28) to the nearest meter, a building 500 meter cover, 140 feet away from the base of the pole and measures the angle of elevation to the top of the pole to be 10°. to the nearest foot, what is the height of the pole? 29) a tree that is 10 yards tall casts a shadow 14 yards long. find the angle of elevation of the sun from the ground to the top of the tree. round to the nearest degree.

Explanation:

Problem 27 (Assuming it's a trigonometry problem about angle of elevation and height)

Step 1: Identify the trigonometric relationship

We have a right triangle where the angle of elevation is \(47^\circ\), the horizontal distance (adjacent side) is 362 meters, and we need to find the height (opposite side) of the tree. We use the tangent function, which is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Let \(h\) be the height of the tree. So, \(\tan(47^\circ)=\frac{h}{362}\).

Step 2: Solve for \(h\)

Multiply both sides by 362: \(h = 362\times\tan(47^\circ)\). We know that \(\tan(47^\circ)\approx1.0724\) (using a calculator). Then \(h\approx362\times1.0724\).

Step 3: Calculate the value

\(362\times1.0724 = 362 + 362\times0.0724=362+26.2088 = 388.2088\approx388\) meters (rounded to a reasonable decimal place).

Step 1: Identify the angle and triangle

The angle of depression is equal to the angle of elevation from the top of the pole to the observer (alternate interior angles). The horizontal distance from the observer to the pole is 140 feet. Let \(h\) be the height of the pole. We use the tangent function: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), where \(\theta = 39^\circ\) (angle of depression/elevation), adjacent side is 140 feet, and opposite side is \(h\).

Step 2: Solve for \(h\)

\(\tan(39^\circ)=\frac{h}{140}\), so \(h = 140\times\tan(39^\circ)\). \(\tan(39^\circ)\approx0.8098\), so \(h\approx140\times0.8098 = 113.372\approx113\) feet (rounded).

Step 1: Identify the trigonometric relationship

We have a right triangle where the length of the shadow (adjacent side) is 14 yards, and we need to find the angle of elevation of the sun. Let the height of the tree be \(h\) (wait, maybe the height is given? Wait, the problem says "A tree that is 14 yard tall casts a shadow 14 yard long". So height (opposite side) \(h = 14\) yards, shadow (adjacent side) \(s = 14\) yards. We use the tangent function: \(\tan(\theta)=\frac{h}{s}\).

Step 2: Calculate the angle

\(\tan(\theta)=\frac{14}{14}=1\). Then \(\theta=\arctan(1)\). We know that \(\arctan(1) = 45^\circ\) (since \(\tan(45^\circ)=1\)).

Answer:

The height of the tree is approximately \(\boldsymbol{388}\) meters (or more precise value depending on calculator precision, e.g., if using more precise \(\tan(47^\circ)\) value).

Problem 28 (Assuming it's a trigonometry problem about angle of depression and height of pole)