Question

topic 2 composing and decomposing functions skills practice continued problem set 3: exploring cubic functions > complete each table. include an expression for the volume. circle the relative maximum or minimum if there is one. 1 height of box (in.) width of box (in.) length of box (in.) volume of box (in.³) 0 8 10 0 1 8 - 2(1) = 6 10 - 2(1) = 8 (1)(6)(8) = 48 1.5 8 - 2(1.5) = 5 10 - 2(1.5) = 7 (1.5)(5)(7) = 52.5 2 8 - 2(2) = 4 10 - 2(2) = 6 (2)(4)(6) = 48 3 8 - 2(3) = 2 10 - 2(3) = 4 (3)(2)(4) = 24 4 8 - 2(4) = 0 10 - 2(4) = 2 (4)(0)(2) = 0 h 8 - 2h 10 - 2h h(8 - 2h)(10 - 2h) 2 radius of cylinder (in.) height of cylinder (in.) base area of cylinder (in.²) volume of cylinder (in.³) 0 0 0 0 0.5 1 2 3 4 r 3r 3.14r²

Explanation:

Step1: Calculate base area and volume for radius = 0.5

• Base area: The formula for the base area of a cylinder is \(A = 3.14r^{2}\). Substitute \(r = 0.5\) into the formula: \(A=3.14\times(0.5)^{2}=3.14\times0.25 = 0.785\)
• Height: Given \(h = 3r\), when \(r = 0.5\), \(h=3\times0.5 = 1.5\)
• Volume: The formula for the volume of a cylinder is \(V=Ah\). Substitute \(A = 0.785\) and \(h = 1.5\) into the formula: \(V=0.785\times1.5=1.1775\)

Step2: Calculate base area and volume for radius = 1

• Base area: Substitute \(r = 1\) into \(A = 3.14r^{2}\). \(A=3.14\times(1)^{2}=3.14\)
• Height: Substitute \(r = 1\) into \(h = 3r\). \(h=3\times1 = 3\)
• Volume: Substitute \(A = 3.14\) and \(h = 3\) into \(V = Ah\). \(V=3.14\times3=9.42\)

Step3: Calculate base area and volume for radius = 2

• Base area: Substitute \(r = 2\) into \(A = 3.14r^{2}\). \(A=3.14\times(2)^{2}=3.14\times4 = 12.56\)
• Height: Substitute \(r = 2\) into \(h = 3r\). \(h=3\times2=6\)
• Volume: Substitute \(A = 12.56\) and \(h = 6\) into \(V = Ah\). \(V=12.56\times6 = 75.36\)

Step4: Calculate base area and volume for radius = 3

• Base area: Substitute \(r = 3\) into \(A = 3.14r^{2}\). \(A=3.14\times(3)^{2}=3.14\times9 = 28.26\)
• Height: Substitute \(r = 3\) into \(h = 3r\). \(h=3\times3 = 9\)
• Volume: Substitute \(A = 28.26\) and \(h = 9\) into \(V = Ah\). \(V=28.26\times9=254.34\)

Step5: Calculate base area and volume for radius = 4

• Base area: Substitute \(r = 4\) into \(A = 3.14r^{2}\). \(A=3.14\times(4)^{2}=3.14\times16 = 50.24\)
• Height: Substitute \(r = 4\) into \(h = 3r\). \(h=3\times4 = 12\)
• Volume: Substitute \(A = 50.24\) and \(h = 12\) into \(V = Ah\). \(V=50.24\times12=602.88\)

Answer:

Radius of Cylinder (in.)	Height of Cylinder (in.)	Base Area of Cylinder (in.²)	Volume of Cylinder (in.³)
0.5	1.5	0.785	1.1775
1	3	3.14	9.42
2	6	12.56	75.36
3	9	28.26	254.34
4	12	50.24	602.88
\(r\)	\(3r\)	\(3.14r^{2}\)	\(9.42r^{3}\) (since \(V=(3.14r^{2})\times(3r)=9.42r^{3}\))