Question

a mall is redesigning a section of the food court that will expand the size of the eating space. this design will be painted on the floor in the new eating area. what is the perimeter of this design? note: this is an equilateral triangle. use \\(\pi = 3.14\\). round the answer to the nearest tenth: \\(\boxed{\text{ft.}}\\)
(image shows an equilateral triangle with side length 30 ft, and three purple circles with diameter 10 ft each, one at each side of the triangle.)

Explanation:

Step1: Analyze the design components

The design has an equilateral triangle and three circular arcs (from the purple circles). The side length of the triangle is 30 ft, and the diameter of each circle is 10 ft, so the radius \( r = \frac{10}{2}=5 \) ft. The three arcs together make up a semicircle? Wait, no, each arc is a portion of a circle. Wait, the triangle is equilateral, so each angle is \( 60^\circ \). Three arcs, each with central angle \( 60^\circ \), so total central angle is \( 3\times60^\circ = 180^\circ \), which is a semicircle. Wait, no, the length of each arc: the formula for arc length is \( \frac{\theta}{360^\circ}\times 2\pi r \). For each arc, \( \theta = 60^\circ \), \( r = 5 \) ft. So each arc length is \( \frac{60^\circ}{360^\circ}\times 2\pi\times5=\frac{1}{6}\times10\pi=\frac{5\pi}{3} \) ft. Three arcs: \( 3\times\frac{5\pi}{3}=5\pi \) ft. Then the straight parts of the triangle: each side of the triangle is 30 ft, but we subtract the diameter of the circle (10 ft) from each side? Wait, no, looking at the diagram, each side of the triangle has a circular arc attached? Wait, maybe the perimeter is the sum of the three straight sides (each 30 - 10? No, wait the triangle has side length 30 ft, and each corner has a circular arc with diameter 10 ft (so radius 5 ft). Wait, maybe the perimeter is the length of the three straight segments (each 30 - 10? No, maybe the straight parts are 30 - 10? Wait, no, the diagram shows the triangle with side 30 ft, and each side has a circle with diameter 10 ft attached. Wait, maybe the perimeter is the sum of the three arc lengths (which make a semicircle) and the three straight segments (each 30 - 10? No, maybe the straight segments are 30 - 10? Wait, no, let's re-examine.

Wait, the problem is to find the perimeter of the design. The design consists of the equilateral triangle's sides, but with circular arcs replacing the corners? Wait, no, the diagram shows a yellow equilateral triangle with three purple circles, each with diameter 10 ft, attached to the sides. So the perimeter of the design would be the sum of the three arc lengths (from the circles) and the three straight parts of the triangle. Wait, each side of the triangle: the length of the straight part is 30 - 10 = 20 ft? Wait, no, the diameter of each circle is 10 ft, so the length of the straight part on each side is 30 - 10 = 20 ft? Wait, no, maybe the arcs are each a 60-degree arc (since the triangle is equilateral, each angle is 60 degrees). So each arc is a 60-degree arc of a circle with radius 5 ft (since diameter is 10 ft, radius 5 ft). So arc length formula: \( \text{arc length} = \frac{\theta}{360^\circ} \times 2\pi r \). For \( \theta = 60^\circ \), \( r = 5 \) ft: \( \frac{60}{360} \times 2\pi \times 5 = \frac{1}{6} \times 10\pi = \frac{5\pi}{3} \) ft. Three arcs: \( 3 \times \frac{5\pi}{3} = 5\pi \) ft. Then the straight parts: each side of the triangle is 30 ft, but we have three straight parts, each 30 - 10 = 20 ft? Wait, no, the diameter of the circle is 10 ft, so the length of the straight segment on each side is 30 - 10 = 20 ft? So three straight segments: \( 3 \times 20 = 60 \) ft. Then total perimeter is \( 60 + 5\pi \). Let's calculate that. \( 5\pi \approx 5 \times 3.14 = 15.7 \) ft. Then total perimeter: \( 60 + 15.7 = 75.7 \) ft? Wait, that doesn't seem right. Wait, maybe the arcs are each a semicircle? No, wait, maybe the three arcs together make a full circle? Wait, no, each arc is 60 degrees, three arcs: 180 degrees, which is a semicircle. Wait, maybe I made a mistake. Let's try again.

Wait, the diameter of each circle is 10 ft, so the circumference of each circle is \( \pi d = 10\pi \) ft. But the arcs: each arc is a 60-degree arc (since the triangle is equilateral, each angle is 60 degrees). So three arcs: 3 * 60 = 180 degrees, which is half the circumference of a circle with diameter 10 ft. So the length of the three arcs is \( \frac{1}{2} \times 10\pi = 5\pi \) ft. Then the straight parts: each side of the triangle is 30 ft, but we subtract the diameter of the circle (10 ft) from each side? Wait, no, each side of the triangle has a circular arc, so the straight part of each side is 30 - 10 = 20 ft? So three straight parts: 3 * 20 = 60 ft. Then total perimeter: 60 + 5\pi. Let's compute that. \( 5\pi \approx 15.7 \), so 60 + 15.7 = 75.7 ft. Wait, but let's check again.

Wait, maybe the perimeter is the sum of the three arc lengths (which are each 60-degree arcs) and the three straight segments (each 30 - 10 = 20 ft). So:

Arc length per arc: \( \frac{60}{360} \times 2\pi \times 5 = \frac{1}{6} \times 10\pi = \frac{5\pi}{3} \)

Three arcs: \( 3 \times \frac{5\pi}{3} = 5\pi \)

Straight segments: each side of the triangle is 30 ft, but we have to subtract the length covered by the circles. Wait, the diameter of each circle is 10 ft, so on each side, the straight part is 30 - 10 = 20 ft. So three straight segments: 3 * 20 = 60 ft.

Total perimeter: 60 + 5\pi ≈ 60 + 15.7 = 75.7 ft.

Wait, but let's confirm. Alternatively, maybe the perimeter is the length of the three arcs (which form a semicircle) plus the three straight sides (each 30 - 10 = 20). So 3*20 + 5\pi = 60 + 15.7 = 75.7.

Step2: Calculate the numerical value

First, calculate the arc length part: three arcs with central angle 60 degrees each. The radius of each circle is \( r = \frac{10}{2} = 5 \) ft.

Arc length formula: \( L = \frac{\theta}{360^\circ} \times 2\pi r \)

For one arc: \( \theta = 60^\circ \), so \( L_1 = \frac{60^\circ}{360^\circ} \times 2\pi \times 5 = \frac{1}{6} \times 10\pi = \frac{5\pi}{3} \) ft

Three arcs: \( L_{\text{arcs}} = 3 \times \frac{5\pi}{3} = 5\pi \) ft

Straight segments: each side of the triangle is 30 ft, but we subtract the diameter of the circle (10 ft) from each side (since the circle is attached to the side, the straight part is 30 - 10 = 20 ft). There are three sides, so \( L_{\text{straight}} = 3 \times 20 = 60 \) ft

Total perimeter: \( L_{\text{total}} = 60 + 5\pi \)

Substitute \( \pi = 3.14 \):

\( 5\pi = 5 \times 3.14 = 15.7 \)

\( L_{\text{total}} = 60 + 15.7 = 75.7 \) ft

Answer:

75.7