You are placing non-slip tape along the perimeter of the bottom of a semicircle-shaped doormat. How much tape will you save appl
1 answer:
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 = 2r
half of the circumference of a circle = r
length of the arc of the semicircle = half of the circumference of a circle = 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 = r + length of the diameter
= × 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 = r + length of the diameter
= × 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.
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Answer:
(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
(B) f(x + h) - f(x) = 8xh + 4h² - 6h
(C)
Step-by-step explanation:
* Lets explain how to solve the problem
- The function f(x) = 4x² - 6x + 6
- To find f(x + h) substitute x in the function by (x + h)
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) = 4(x + h)² - 6(x + h) + 6
- Lets simplify 4(x + h)²
∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²
∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²
- Lets simplify 6(x + h)
∵ 6(x + h) = 6(x) + 6(h)
∴ 6(x + h) = 6x + 6h
∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6
- Remember (-)(+) = (-)
∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
- Lets find f(x + h) - f(x)
∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)
- Remember (-)(-) = (+)
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6
- Simplify by adding the like terms
∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)
∴ f(x + h) - f(x) = 8xh + 4h² - 6h
* (B) f(x + h) - f(x) = 8xh + 4h² - 6h
- Lets find
∵ f(x + h) - f(x) = 8xh + 4h² - 6h
∴
- Simplify by separate the three terms
∴
∴
* (C)