Question

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juan is clearing land in the shape of a circle to plant a new tree. the diameter of the space he needs to clear is 52 inches. by midday, he has cleared a sector of the land cut off by a central angle of 140°. what is the arc length and the area of land he has cleared by midday?
the land juan has cleared by midday has an arc length of about inches and an area of about square inches.

Explanation:

Step1: Find radius of the circle

The diameter is 52 inches, so radius $r = \frac{52}{2} = 26$ inches.

Step2: Calculate arc length

Arc length formula: $L = \frac{\theta}{360^\circ} \times 2\pi r$, where $\theta=140^\circ$.
$L = \frac{140^\circ}{360^\circ} \times 2\pi \times 26 = \frac{7}{18} \times 52\pi \approx \frac{7}{18} \times 163.36 \approx 63.07 \approx 64$ inches.

Step3: Calculate sector area

Sector area formula: $A = \frac{\theta}{360^\circ} \times \pi r^2$
$A = \frac{140^\circ}{360^\circ} \times \pi \times 26^2 = \frac{7}{18} \times 676\pi \approx \frac{7}{18} \times 2123.72 \approx 820.07 \approx 826$ square inches.

Answer:

The land Juan has cleared by midday has an arc length of about 64 inches and an area of about 826 square inches.