5 Steps to a 5 AP Calculus BC 2019

158 STEP 4. Review the Knowledge You Need to Score High

Example Volume Problem (with calculator)

If an open box is to be made using a square sheet of tin, 20 inches by 20 inches, by cutting

a square from each corner and folding the sides up, find the length of a side of the square

being cut so that the box will have a maximum volume.

Solution:

Step 1: Draw a diagram. (See Figure 8.2-3.)

xx

x

x

x

x

x

x

x

20

20–2x

20–2x

20–2

x

20

20–2x

Figure 8.2-3

Step 2: Letxbe the length of a side of the square to be cut from each corner.

Step 3: The volume of the box isV(x)=x(20− 2 x)(20− 2 x).

The domain ofVis [0, 10].

Step 4: DifferentiateV(x).

Enterd(x∗(20− 2 x)∗(20− 2 x),x) and we have 4(x−10)(3x−10).

Step 5: V′(x) is defined for all real numbers:

SetV′(x)=0 by entering: [Solve] (4(x−10)(3x−10)=0,x), and obtainx= 10

orx=

10

3

. The critical numbers ofV(x) arex=10 andx=

10

3

.V(10)=0 and

V

(

10

3

)

= 592 .59. SinceV(10)=0, you need to test onlyx=

10

3

.