Question

1. you need a piece of steel cut to a certain shape to fit on a machine. the shape is given below

find the area of this piece of steel.
if the thickness of the steel is 1/8 inch, what is
the volume of the piece?
steel weighs 0.283 pounds per cubic inch. how many pounds is this piece?

Explanation:

Step1: Calculate the area of the composite shape

• Divide the shape into two rectangles.
• For the first rectangle with dimensions \(18\times(6 + 2+7 + 3)\):
• \(6+2 + 7+3=18\)
• Area \(A_1=18\times18 = 324\) square inches.
• For the second rectangle (the cut - out part) with dimensions \(7\times(18-(2 + 2))\):
• \(18-(2 + 2)=14\)
• Area \(A_2=7\times14=98\) square inches.
• The area of the steel piece \(A = A_1 - A_2\)
• \(A=324-98 = 226\) square inches.

Step2: Calculate the volume of the steel piece

• The formula for the volume of a rectangular - prism - like shape (since thickness is uniform) is \(V=A\times h\), where \(h\) is the thickness.
• Given \(h=\frac{1}{8}\) inch and \(A = 226\) square inches.
• \(V=226\times\frac{1}{8}=\frac{226}{8}=28.25\) cubic inches.

Step3: Calculate the weight of the steel piece

• Given the weight per cubic inch is \(0.283\) pounds.
• Use the formula \(W=V\times

ho\), where \(
ho\) is the weight per unit volume.

• \(W = 28.25\times0.283\)
• \(W=28.25\times(0.2 + 0.08+0.003)\)
• \(W=(28.25\times0.2)+(28.25\times0.08)+(28.25\times0.003)\)
• \(28.25\times0.2 = 5.65\), \(28.25\times0.08=2.26\), \(28.25\times0.003 = 0.08475\)
• \(W=5.65+2.26 + 0.08475=7.99475\approx8.0\) pounds

Answer:

1. The area of the steel piece is \(226\) square inches.
2. The volume of the steel piece is \(28.25\) cubic inches.
3. The weight of the steel piece is approximately \(8.0\) pounds.