To find the x-component of a vector when you have the magnitude of the vector and its y-component, you can use the Pythagorean theorem. Here’s a detailed, step-by-step solution:
1. Identify the given values:
- Magnitude of the vector, [tex]\( \text{magnitude} = 43 \, \text{meters} \)[/tex]
- Y-component of the vector, [tex]\( y = 30 \, \text{meters} \)[/tex]
2. Recall the relationship between the components and the magnitude of the vector:
The magnitude of the vector (often denoted as [tex]\( r \)[/tex]) relates to the x and y components through the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
3. Rearrange the formula to solve for the x-component:
[tex]\[
x^2 = r^2 - y^2
\][/tex]
[tex]\[
x = \sqrt{r^2 - y^2}
\][/tex]
4. Substitute the given values into the equation:
- [tex]\( r = 43 \, \text{meters} \)[/tex]
- [tex]\( y = 30 \, \text{meters} \)[/tex]
[tex]\[
x = \sqrt{43^2 - 30^2}
\][/tex]
5. Calculate the squares:
- [tex]\( 43^2 = 1849 \)[/tex]
- [tex]\( 30^2 = 900 \)[/tex]
6. Find the difference:
[tex]\[
1849 - 900 = 949
\][/tex]
7. Take the square root of the result:
[tex]\[
x = \sqrt{949}
\][/tex]
8. Approximate the square root:
The x-component of the vector [tex]\( x \approx 30.81 \, \text{meters} \)[/tex].
Therefore, the x-component of the vector is approximately [tex]\( 30.81 \, \text{meters} \)[/tex].
