Answer:
The amount of tape to be saved is 15.43 inches.
Step-by-step explanation:
Semicircle is half of a given circle, consisting of an arc and a diameter. Its perimeter is determined by adding the length of its arc to the diameter.
perimeter of a semicircle = length of its arc + its diameter
length of the arc = half of the circumference of its circle
circumference of a circle = 2[tex]\pi[/tex]r
half of the circumference of a circle = [tex]\pi[/tex]r
length of the arc of the semicircle = half of the circumference of a circle = [tex]\pi[/tex]r
The initial diameter of the semicircle = 30 inches
⇒ its radius, r = 15 inches
Amount of tape to be saved = Initial perimeter of the semicircle - perimeter of the inside of the semicircle
Initial perimeter of the semicircle = [tex]\pi[/tex]r + length of the diameter
= [tex]\frac{22}{7}[/tex] × 15 + 30
= 47.1429 + 30
= 77.1429 inches
Initial perimeter of the semicircle is 77.1429 inches.
The inside semicircle is formed, since a tape of width 3 inches has been placed along the perimeter of the initial doormat. Then,
diameter of the inside semicircle = 30 - 6
= 24 inches
⇒ its radius = 12 inches
perimeter of the inside semicircle = [tex]\pi[/tex]r + length of the diameter
= [tex]\frac{22}{7}[/tex] × 12 + 24
= 37.7143 + 24
= 61.7143 inches
perimeter of the inside semicircle is 61.7143 inches.
Amount of tape to be saved = 77.1429 - 61.7143
= 15.4286
The amount of tape to be saved is 15.43 inches.
