To guard against a fall, a ladder should make an angle of 75o or less with the ground. What is the maximum height that a 20-foot ladder can reach safely to the nearest tenth of a foot?
in my diagram I set it up using the sin function since I want to find the height so I did (sin75=x/20) giving me 19.3
I think its 19.31
sin(75)=x/20
20 x sin(75) = 19.31
Well, looks like you've got it all figured out! Using the sin function to find the height was a smart move. With an angle of 75 degrees and a ladder length of 20 feet, your calculation gives you a maximum height of 19.3 feet. But hey, don't worry, that height is still pretty high. Just make sure you bring a tall friend along in case you need someone to catch you if you stumble! Safety first, after all.
To find the maximum height that a 20-foot ladder can reach safely, we can use trigonometry. By applying the sine function, as you did, we can solve for the height (opposite side).
The equation you used, sin75 = x/20, is correct. However, the value you obtained, 19.3, seems to be a rounding error. Let's go through the calculations step-by-step to ensure accuracy:
1. Start with the equation: sin75 = x/20.
2. Multiply both sides by 20 to isolate the x: 20 * sin75 = x.
3. Use a calculator to find the sine of 75 degrees: sin75 = 0.9659 (rounded to four decimal places).
4. Multiply 20 by 0.9659: 20 * 0.9659 = 19.318 (rounded to three decimal places).
Therefore, the maximum height that a 20-foot ladder can reach safely, to the nearest tenth of a foot, is 19.3 feet.
Your approach using the sine function is correct! Let's go through the steps to solve the problem and find the maximum height reached by a 20-foot ladder when it makes an angle of 75 degrees or less with the ground.
1. Define the given information:
- Ladder length = 20 feet
- Maximum allowed angle = 75 degrees
2. Use the sine function (sin) and form the equation:
The sine function relates the angle of a right triangle to the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height the ladder reaches, and the hypotenuse is the length of the ladder itself.
The equation using the sine function is: sin(angle) = opposite/hypotenuse
Plugging in the values: sin(75) = x/20
3. Solve for x:
To isolate x, multiply both sides of the equation by 20:
20 * sin(75) = x
4. Calculate x:
Using a calculator or trigonometric table, find the value of sin(75) ≈ 0.9659
Multiply 20 by sin(75):
20 * 0.9659 ≈ 19.318
5. Round the answer to the nearest tenth of a foot:
The maximum height reached by the ladder is approximately 19.3 feet (to the nearest tenth of a foot).
So, your calculation of 19.3 feet as the maximum safe height for a 20-foot ladder when making an angle of 75 degrees or less with the ground is correct!