A drawbridge is l = 110 feet long when stretched across a river. As shown in the figure below, the two sections of the bridge can be rotated upward through an angle of α = 34°.

a)If the water level is 35 feet below the closed bridge, find the distance d between the end of a section and the water level when the bridge is fully open. (Round your answer to one decimal place.)
b)Approximately how far apart are the ends of the two sections when the bridge is fully opened, as shown in the figure? (Round your answer to one decimal place.)

a) Well, let's first calculate the height of the bridge when fully open. We can use a bit of trigonometry here. Since the bridge forms a right triangle with the water level, we can use the cosine function:

cos(α) = Adjacent / Hypotenuse

cos(34°) = d / 110

Now, let's solve for d:

d = 110 * cos(34°)

d ≈ 91.5 feet

So, the distance between the end of a section and the water level when the bridge is fully open is approximately 91.5 feet.

b) When the bridge is fully opened, we have two congruent right triangles formed by the bridge's sections and the distance between their ends. The hypotenuse of each triangle is 110 feet (the length of the bridge).

To find the distance between the ends of the two sections, we need to calculate twice the base (or twice the adjacent side of one of the triangles) using the sine function:

sin(α) = Opposite / Hypotenuse

sin(34°) = 2x / 110

Let's solve for 2x:

2x = 110 * sin(34°)

2x ≈ 63.3 feet

So, approximately, the distance between the ends of the two sections when the bridge is fully opened is around 63.3 feet. Now that's quite a gap! Just make sure not to drop anything in there, or it will end up in the drink.

To solve these questions, we can use the properties of a right triangle and trigonometric functions.

a) To find the distance d between the end of a section and the water level when the bridge is fully open, we can use the sine function. The opposite side of the right triangle formed is the height of the water level, which is 35 feet. The hypotenuse of the right triangle is the length of the bridge, which is 110 feet. The sine of the angle α is equal to the ratio of the opposite side to the hypotenuse.

sin(α) = opposite/hypotenuse

sin(34°) = 35/d

To find d, we can rearrange the equation:

d = 35 / sin(34°)

Using a calculator, the value of d is approximately 64.4 feet (rounded to one decimal place).

Therefore, the distance between the end of a section and the water level when the bridge is fully open is approximately 64.4 feet.

b) To find approximately how far apart the ends of the two sections are when the bridge is fully opened, we can use the cosine function. The adjacent side of the right triangle formed is the distance between the ends of the two sections (let's call it x). The hypotenuse of the right triangle is still the length of the bridge, which is 110 feet. The cosine of the angle α is equal to the ratio of the adjacent side to the hypotenuse.

cos(α) = adjacent/hypotenuse

cos(34°) = x/110

To find x, we can rearrange the equation:

x = 110 * cos(34°)

Using a calculator, the value of x is approximately 91.2 feet (rounded to one decimal place).

Therefore, approximately, the ends of the two sections are about 91.2 feet apart when the bridge is fully opened.

To solve this problem, we can use trigonometry. Let's break down the problem step by step:

a) Finding the distance d between the end of a section and the water level when the bridge is fully open:

1. Let's consider the triangle formed by the bridge, the distance d, and the water level. This triangle is a right triangle.

2. The hypotenuse of this right triangle is the length of the bridge, l, which is given as 110 feet.

3. The angle between the bridge and the water level, α, is given as 34°.

4. We can use the trigonometric function cosine (cos) to relate the adjacent side (d) to the hypotenuse (l) and the angle (α) using the cosine ratio: cos(α) = adjacent/hypotenuse.

5. Rearranging the equation to solve for the adjacent side, we have: adjacent = cos(α) * hypotenuse.

6. Substituting the values, we have: d = cos(34°) * 110.

7. Using a calculator, we find that cos(34°) is approximately 0.829, so: d = 0.829 * 110.

8. Calculating this, we get: d ≈ 91.2 feet (rounded to one decimal place).

Therefore, the distance between the end of a section and the water level when the bridge is fully open is approximately 91.2 feet.

b) Approximating the distance between the ends of the two sections when the bridge is fully opened:

1. When the bridge is fully open, both sections are rotated upward by the same angle, α, which is given as 34°.

2. The distance between the ends of the two sections can be found by calculating twice the side adjacent to the angle α in a right triangle.

3. Using the same approach as before, we can calculate this distance using the cosine ratio: adjacent = cos(α) * hypotenuse.

4. Substituting the values, we have: distance = 2 * cos(34°) * l.

5. Calculating this, we get: distance = 2 * 0.829 * 110.

6. Simplifying this, we have: distance ≈ 182.4 feet (rounded to one decimal place).

Therefore, the approximate distance between the ends of the two sections when the bridge is fully opened is approximately 182.4 feet.

Draw a diagram and review your basic trig functions. You will see that the height the section has been raised is

h/55 = sinα

The horizontal distance from the bank is x = 55cosα

So, the end has moved 55(1-cosα) ft closer to the bank.

Now you can answer the questions.