Question #287009
  1. a metal washer 1 inch. in diameter is pierced by a 1/2 - inch hole. what is the area of one of the washer?
  2. the vertical end of a trough has the following dimensions width at top 4.4ft., width at bottom 3.2ft., depth 3.5ft. find the area of the end of the trough.
  3. the surface of hemispherical dome whose diameter is 36ft. is to be covered with gold leaf which costa 15 cents per square inch. what must be paid for the golf leaf?
  4. a cubic foot of ivory weighs 114Ib. find the weight of 1000 ivory billiard balls 21 in. in diameter?

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1
Expert's answer
2022-01-16T17:11:59-0500

1.



A metal washer is made from two concentric circles and its a ring shape. 


Let r1=r_1= the radius of inner circle

      r2=r_2= the radius of outer circle

 Surface are


A=πr22πr12A=\pi r_2^2-\pi r_1^2

Given 2r1=12in,2r2=1in.2r_1=\dfrac{1}{2}in, 2r_2=1 in.

Then


r1=14in,r2=12inr_1=\dfrac{1}{4}in, r_2=\dfrac{1}{2}in

A=π(12in)2π(14in)2=3π16in2A=\pi(\dfrac{1}{2}in)^2-\pi(\dfrac{1}{4}in)^2=\dfrac{3\pi}{16}in^2

The area of one of the washer is 3π16 in2.\dfrac{3\pi}{16}\ in^2.


2. From the description of the given word problem, the vertical end of a trough is a trapezoid. 



Given a=4.4ft,b=3.2ft,h=3.5ft.a=4.4ft, b=3.2ft, h=3.5ft.

The area of a trapezoid is A=a+b2h.A=\dfrac{a+b}{2}h.

Substitute


A=4.4ft+3.2ft2(3.5ft)=13.3ft2A=\dfrac{4.4ft+3.2ft}{2}(3.5ft)=13.3ft^2

The area of the end of the trough is 13.3ft2.13.3ft^2.


3. The curved surface area of the dome is


A=12(4πr2)=πd22A=\dfrac{1}{2}(4\pi r^2)=\dfrac{\pi d^2}{2}

Given d=36ft=432in.d=36ft=432in. Then


π(432in)22=93312π in2\dfrac{\pi (432in)^2}{2}=93 312\pi\ in^2

What must be paid for the golf leaf?


93312π in2(0.15$/in2)=$43972.2593 312\pi\ in^2(0.15\$/in^2)=\$43972.25

4. Volume of one billiard ball


V1=4π3r3=πd36V_1=\dfrac{4\pi}{3}r^3=\dfrac{\pi d^3}{6}

Given d=21in=1.75ft.d=21in=1.75ft. Then the volume of 1000 ivory billiard balls is


V=1000π(1.75ft)36=5359.375π6 ft3V=1000\cdot\dfrac{\pi(1.75ft)^3}{6}=\dfrac{5359.375\pi}{6}\ ft^3

The weight (mass) of 1000 ivory billiard balls is


M=ρVM=\rho V

M=114lb/ft3(5359.375π6 ft3)M=114lb/ft^3 \cdot(\dfrac{5359.375\pi}{6}\ ft^3)

=319902.5 lb=145105.21 kg=319902.5\ lb=145105.21\ kg


The weight (mass) of 1000 ivory billiard balls is 319902.5 lb319902.5\ lb or 145105.21 kg.145105.21\ kg.



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