HOW CAN I DO THIS WITHOUT A CALCULATOR?
A giant redwood tree casts a shadow, as shown in the figure. (Hint: tan25.7 = 0.4817)
(A) If the shadow casted is 532 At long, and if the angle of elevation of the sun is 25.7°, then the height of the tree
approximately equals to 378 ft.
(B) If the angle of elevation of the sun is 48.5°, the height of the tree is 256 ft, then the shadow casted is approximately
312 ft.
(C) If the angle of elevation of the sun is 36.8°, the shadow casted is 312 ft then the height of the tree is approximately
432 ft.
(D) None of the above (A) (B) (C) is true
I don't know how good you are at doing division in your head, but in each case,
height/shadow = tanθ
In the case of (A), since 0.48 is about 1/2, the tree will be a bit less than 1/2 as long as the shadow. Since 378/532 > 1/2 I guess (A) is false.
since tan 45° = 1, B is false using similar reasoning
Same for C.
So, I guess D is the only choice
To solve this problem without using a calculator, we can use basic trigonometric ratios such as tangent.
(A) Let's start with option (A). We are given the shadow length (532 ft) and the angle of elevation (25.7°).
Using the tangent ratio, we can say that tan(25.7°) = height of the tree / shadow length.
From the hint, we know that tan(25.7°) is approximately 0.4817.
Now we can solve for the height of the tree:
height of the tree = 0.4817 x 532 ft ≈ 256 ft.
(B) Moving on to option (B), we are given the angle of elevation (48.5°) and the height of the tree (256 ft).
Using the same tangent ratio as before, we can say that tan(48.5°) = height of the tree / shadow length.
Solving for the shadow length:
shadow length = height of the tree / tan(48.5°).
Plugging in the values:
shadow length ≈ 256 ft / tan(48.5°).
Without a calculator, we cannot find the exact value, so we can estimate it to be approximately 312 ft.
(C) Lastly, in option (C), we are given the angle of elevation (36.8°) and the shadow length (312 ft).
Using the same tangent ratio, we can say that tan(36.8°) = height of the tree / shadow length.
Solving for the height of the tree:
height of the tree = tan(36.8°) x shadow length.
Without a calculator, we cannot find the exact value, so we can estimate it to be approximately 432 ft.
(D) None of the above (A)(B)(C) is true, as option (C) matches the given information.
Therefore, the correct answer is option (D) None of the above (A)(B)(C) is true.
To solve this problem without a calculator, we can use trigonometric ratios and basic arithmetic.
(A) To find the height of the tree, we can use the tangent ratio. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tree and the adjacent side is the length of the shadow.
We are given the angle of elevation of the sun, which is 25.7°, and the length of the shadow, which is 532 ft. The tangent of 25.7° is given as 0.4817. We can set up the equation:
height of tree / length of shadow = tangent(angle of elevation)
height of tree / 532 = 0.4817
To find the height of the tree, we multiply both sides of the equation by 532:
height of tree = 0.4817 * 532
height of tree ≈ 378 ft
Therefore, option (A) is true.
(B) Using the same approach, we can solve for the length of the shadow. We are given the angle of elevation of the sun, which is 48.5°, and the height of the tree, which is 256 ft.
Setting up the equation:
height of tree / length of shadow = tangent(angle of elevation)
256 / length of shadow = tangent(48.5°)
Rearranging the equation to solve for the length of the shadow:
length of shadow = height of tree / tangent(angle of elevation)
length of shadow = 256 / tangent(48.5°)
Since we don't have the tangent value for 48.5°, we cannot calculate the exact length of the shadow without a calculator. Therefore, option (B) cannot be determined without a calculator.
(C) This question provides the angle of elevation of the sun as 36.8° and the length of the shadow as 312 ft. To find the height of the tree, we use the same approach as in question (A).
height of tree / length of shadow = tangent(angle of elevation)
height of tree / 312 = tangent(36.8°)
To solve for the height of the tree, multiply both sides of the equation by 312:
height of tree = 312 * tangent(36.8°)
Again, since we don't have the tangent value for 36.8°, we cannot calculate the exact height of the tree without a calculator. Therefore, option (C) cannot be determined without a calculator.
(D) Based on the information provided, we can conclude that both options (B) and (C) cannot be determined without a calculator. Thus, the correct answer is option (D).