Question
A stone is suspended from a point, P, by a piece of string 50cm long. It swings backward and forward. Calculate the angle the string makes with the vertical when the stone is 35cm vertically below P.
Answers
GPT-4o mini
To find the angle that the string makes with the vertical when the stone is 35 cm vertically below point P, we can use some basic trigonometry.
1. **Identify the lengths:**
- Length of the string (L) = 50 cm
- Vertical distance from point P to the stone (h) = 35 cm
2. **Find the horizontal distance (x) from the vertical line:**
- The vertical distance from point P to the bottom of the swing (the lowest point the stone can reach) is 50 cm when the stone is directly below point P.
- When the stone is 35 cm directly below P, the vertical distance to point P is less, leaving a vertical distance that can be used to calculate the horizontal component.
The vertical distance of the string from point P to the stone is:
\[
V = L - h = 50 \text{ cm} - 35 \text{ cm} = 15 \text{ cm}
\]
3. **Use the Pythagorean theorem:**
In the right triangle formed by the string, the vertical line to point P, and the horizontal distance from the vertical line to the position of the stone:
\[
L^2 = V^2 + x^2
\]
where
- \(L = 50 \text{ cm}\)
- \(V = 15 \text{ cm}\)
- \(x\) is the horizontal distance.
Plugging in the values:
\[
50^2 = 15^2 + x^2
\]
\[
2500 = 225 + x^2
\]
\[
x^2 = 2500 - 225 = 2275
\]
\[
x = \sqrt{2275} \approx 47.7 \text{ cm}
\]
4. **Calculate the angle (θ) with respect to the vertical:**
To find the angle θ that the string makes with the vertical, we use:
\[
\cos(θ) = \frac{V}{L} = \frac{15}{50} = 0.3
\]
Now, solving for the angle:
\[
θ = \cos^{-1}(0.3)
\]
5. **Find the angle value:**
We can use a calculator to determine:
\[
θ \approx 72.34^\circ
\]
Therefore, the angle the string makes with the vertical when the stone is 35 cm vertically below point P is approximately \(72.34^\circ\).
1. **Identify the lengths:**
- Length of the string (L) = 50 cm
- Vertical distance from point P to the stone (h) = 35 cm
2. **Find the horizontal distance (x) from the vertical line:**
- The vertical distance from point P to the bottom of the swing (the lowest point the stone can reach) is 50 cm when the stone is directly below point P.
- When the stone is 35 cm directly below P, the vertical distance to point P is less, leaving a vertical distance that can be used to calculate the horizontal component.
The vertical distance of the string from point P to the stone is:
\[
V = L - h = 50 \text{ cm} - 35 \text{ cm} = 15 \text{ cm}
\]
3. **Use the Pythagorean theorem:**
In the right triangle formed by the string, the vertical line to point P, and the horizontal distance from the vertical line to the position of the stone:
\[
L^2 = V^2 + x^2
\]
where
- \(L = 50 \text{ cm}\)
- \(V = 15 \text{ cm}\)
- \(x\) is the horizontal distance.
Plugging in the values:
\[
50^2 = 15^2 + x^2
\]
\[
2500 = 225 + x^2
\]
\[
x^2 = 2500 - 225 = 2275
\]
\[
x = \sqrt{2275} \approx 47.7 \text{ cm}
\]
4. **Calculate the angle (θ) with respect to the vertical:**
To find the angle θ that the string makes with the vertical, we use:
\[
\cos(θ) = \frac{V}{L} = \frac{15}{50} = 0.3
\]
Now, solving for the angle:
\[
θ = \cos^{-1}(0.3)
\]
5. **Find the angle value:**
We can use a calculator to determine:
\[
θ \approx 72.34^\circ
\]
Therefore, the angle the string makes with the vertical when the stone is 35 cm vertically below point P is approximately \(72.34^\circ\).