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Sagot :
### Question 5:
To determine how far the ant would travel if it walked straight down the side of the tower until it reached the ground, we need to consider the structure of the problem geometrically.
Given:
- The base of the tower: 10 meters
- The height of the tower: 20 meters
To solve this, we can use a trigonometric ratio. However, in this specific scenario, a more direct approach would be the Pythagorean theorem, which relates the sides of a right triangle for our problem.
The trigonometric ratio best suited in this context would be the tangent (tan), which is the ratio of the opposite side (height) to the adjacent side (base). However, since we have both perpendicular legs, using the Pythagorean theorem:
### Step-by-Step Solution:
1. Identify the sides of the right triangle:
- [tex]\( a = 10 \)[/tex] meters (base of the tower)
- [tex]\( b = 20 \)[/tex] meters (height of the tower)
2. Using the Pythagorean theorem, [tex]\( a^2 + b^2 = c^2 \)[/tex]:
- Substitute the values:
[tex]\[ 10^2 + 20^2 = c^2 \][/tex]
- Calculate the squares:
[tex]\[ 100 + 400 = c^2 \][/tex]
- Sum the squares:
[tex]\[ 500 = c^2 \][/tex]
- Find the hypotenuse [tex]\( c \)[/tex] by taking the square root:
[tex]\[ c = \sqrt{500} \][/tex]
3. Simplify the square root:
[tex]\[ c \approx 22.36 \, \text{meters} \][/tex]
Thus, the ant would travel approximately 22.36 meters.
### Question 6:
To confirm our answer using the Pythagorean theorem, we'll walk through the same process:
1. Formulate the problem using the Pythagorean theorem:
- [tex]\( a^2 + b^2 = c^2 \)[/tex] where [tex]\( a \)[/tex] is the base, [tex]\( b \)[/tex] is the height, and [tex]\( c \)[/tex] is the hypotenuse.
2. Plug in the values:
[tex]\[ 10^2 + 20^2 = c^2 \][/tex]
3. Perform the calculations:
[tex]\[ 100 + 400 = c^2 \][/tex]
4. Sum the squares:
[tex]\[ 500 = c^2 \][/tex]
5. Compute the hypotenuse:
[tex]\[ c = \sqrt{500} \][/tex]
[tex]\[ c \approx 22.36 \][/tex]
From this, we confirm that the distance the ant travels, as calculated using the Pythagorean theorem, is indeed approximately 22.36 meters.
### Summary:
- Method: Pythagorean theorem (1 point)
- Shown work: Calculations detailed (2 points)
- Final Answer: Approximately 22.36 meters (1 point).
### Confirmation:
- Reapplied the Pythagorean theorem yielding the same result of approximately 22.36 meters (3 points).
Thus, through both the trigonometric approach and verification with the Pythagorean theorem, we have confirmed the correct answer is approximately 22.36 meters.
To determine how far the ant would travel if it walked straight down the side of the tower until it reached the ground, we need to consider the structure of the problem geometrically.
Given:
- The base of the tower: 10 meters
- The height of the tower: 20 meters
To solve this, we can use a trigonometric ratio. However, in this specific scenario, a more direct approach would be the Pythagorean theorem, which relates the sides of a right triangle for our problem.
The trigonometric ratio best suited in this context would be the tangent (tan), which is the ratio of the opposite side (height) to the adjacent side (base). However, since we have both perpendicular legs, using the Pythagorean theorem:
### Step-by-Step Solution:
1. Identify the sides of the right triangle:
- [tex]\( a = 10 \)[/tex] meters (base of the tower)
- [tex]\( b = 20 \)[/tex] meters (height of the tower)
2. Using the Pythagorean theorem, [tex]\( a^2 + b^2 = c^2 \)[/tex]:
- Substitute the values:
[tex]\[ 10^2 + 20^2 = c^2 \][/tex]
- Calculate the squares:
[tex]\[ 100 + 400 = c^2 \][/tex]
- Sum the squares:
[tex]\[ 500 = c^2 \][/tex]
- Find the hypotenuse [tex]\( c \)[/tex] by taking the square root:
[tex]\[ c = \sqrt{500} \][/tex]
3. Simplify the square root:
[tex]\[ c \approx 22.36 \, \text{meters} \][/tex]
Thus, the ant would travel approximately 22.36 meters.
### Question 6:
To confirm our answer using the Pythagorean theorem, we'll walk through the same process:
1. Formulate the problem using the Pythagorean theorem:
- [tex]\( a^2 + b^2 = c^2 \)[/tex] where [tex]\( a \)[/tex] is the base, [tex]\( b \)[/tex] is the height, and [tex]\( c \)[/tex] is the hypotenuse.
2. Plug in the values:
[tex]\[ 10^2 + 20^2 = c^2 \][/tex]
3. Perform the calculations:
[tex]\[ 100 + 400 = c^2 \][/tex]
4. Sum the squares:
[tex]\[ 500 = c^2 \][/tex]
5. Compute the hypotenuse:
[tex]\[ c = \sqrt{500} \][/tex]
[tex]\[ c \approx 22.36 \][/tex]
From this, we confirm that the distance the ant travels, as calculated using the Pythagorean theorem, is indeed approximately 22.36 meters.
### Summary:
- Method: Pythagorean theorem (1 point)
- Shown work: Calculations detailed (2 points)
- Final Answer: Approximately 22.36 meters (1 point).
### Confirmation:
- Reapplied the Pythagorean theorem yielding the same result of approximately 22.36 meters (3 points).
Thus, through both the trigonometric approach and verification with the Pythagorean theorem, we have confirmed the correct answer is approximately 22.36 meters.
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