To determine the length of the minor arc [tex]\(XZ\)[/tex] in circle [tex]\(Y\)[/tex] with a given central angle [tex]\(X Y Z\)[/tex] of [tex]\(70^{\circ}\)[/tex] and a radius of [tex]\(3\)[/tex] meters, follow these steps:
1. Calculate the circumference of the entire circle:
The formula to find the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[
C = 2 \times \pi \times \text{radius}
\][/tex]
Substituting the given radius [tex]\(r\)[/tex] of [tex]\(3\)[/tex] meters:
[tex]\[
C = 2 \times \pi \times 3 = 6\pi \approx 18.84955592153876 \text{ meters}
\][/tex]
2. Determine the fraction of the circle represented by the central angle:
Since the central angle is [tex]\(70^{\circ}\)[/tex] and a full circle is [tex]\(360^{\circ}\)[/tex]:
[tex]\[
\text{Fraction} = \frac{70}{360} = \frac{7}{36} \approx 0.19444444444444445
\][/tex]
3. Calculate the length of the minor arc [tex]\(XZ\)[/tex]:
The length of the arc is the circumference multiplied by the fraction of the circle represented by the angle:
[tex]\[
\text{Arc length} = C \times \text{Fraction}
\][/tex]
Substituting the values:
[tex]\[
\text{Arc length} \approx 18.84955592153876 \times 0.19444444444444445 \approx 3.7 \text{ meters}
\][/tex]
Hence, the approximate length of the minor arc [tex]\(XZ\)[/tex], rounded to the nearest tenth of a meter, is:
[tex]\[
\boxed{3.7 \text{ meters}}
\][/tex]
