Problem 28 - The Calculus with Analytic Geometry

Question

A wooden box is to be made of lumber that is \(\frac{2}{3}\) in. thick. The inside length is to be \(6 \mathrm{ft}\), the inside width is to be \(3 \mathrm{ft}\), the inside depth is to be \(4 \mathrm{ft}\), and the box is to have no top. Use the total differential to find the approximate amount of lumber to be used in the box.

Step-by-Step Solution

Answer

Box needs about 58.67 sq ft.

Step 1: Understand the Dimensions

The internal dimensions of the box are given: length = 6 ft, width = 3 ft, and depth = 4 ft. Remember, the box has no top.

Step 2: Calculate the Outer Dimensions

The lumber thickness is \( \frac{2}{3} \) inches or \( \frac{1}{18} \) feet. Add twice the thickness for each internal dimension to find the outer dimensions. Outer length = 6 + 2 \( \times \frac{1}{18} \) feet, Outer width = 3 + 2 \( \times \frac{1}{18} \) feet, Outer depth = 4 + \( \frac{1}{18} \) feet.

Step 3: Compute the Outer Dimensions

Using the formula, Outer length = \( 6 + \frac{1}{9} = \frac{55}{9} \) ft, Outer width = \( 3 + \frac{1}{9} = \frac{28}{9} \) ft, Outer depth = \( 4 + \frac{1}{18} = \frac{73}{18} \) ft.

Step 4: Calculate the Surface Area

Calculate the surface area (SA) of the outer box: \[ SA = 2lw + 2lh + lw \] where l= \( \frac{55}{9} \), w= \( \frac{28}{9} \), and h= \( \frac{73}{18} \). Thus, SA = 2 \( \times \frac{55}{9} \times \frac{28}{9} \) + 2 \( \times \frac{55}{9} \times \frac{73}{18} \) + \( \times \frac{28}{9} \times \frac{73}{18} \).

Step 5: Calculate the Amount of Lumber Required

Subtract the inner surface area from the outer: the inner dimensions are provided, hence compute the inner SA and subtract from outer SA. Inner SA = 2(6)(3) + 2(6)(4) + 2(3)(4) - 1(6)(3).

Step 6: Calculate and Summarize

Compute the exact values of all surface areas to find the total wood required.

Key Concepts

Differential Calculus

Differential calculus is all about understanding how things change. It's a powerful tool in mathematics used to find rates of change. In this problem, we used the total differential to approximate the amount of lumber required for the wooden box. We start by calculating the box's inner dimensions and then consider changes in each dimension due to the lumber's thickness. This change is small, so using the total differential helps us find an approximate value without complex calculations. Basically, differential calculus lets us capture small changes efficiently, making our calculations more manageable.

Surface Area Calculation

Surface area calculations involve finding the total area covering a 3D object. For this wooden box, we need to determine both the inner and outer surface areas. The box has no top, so our calculations focus on the sides and bottom. We use the dimensions given and the thickness of the lumber to find the outer dimensions. Then, we apply the surface area formula for a rectangular prism:

• 2lw (two sides)
• 2lh (front and back)
• wj (bottom)

Plugging in the lengths, widths, and heights from both the inner and outer dimensions gives us the total surface area of the box, critical for finding how much lumber is needed.

Geometry in Calculus

Geometry in calculus connects shapes' properties with calculus principles. In this exercise, we deal with a rectangular prism (the wooden box). Understanding the geometric shape helps to set up our calculus problem accurately. We calculate the outer dimensions by adding the thickness of the wood to our inner measurements. Then, by understanding the shape of the box, we determine which surfaces need to be taken into account for our calculations—especially noting that the top is missing. This geometrical insight, combined with our calculus tools, helps us accurately figure out the required amount of lumber.

Volume and Surface Area

Volume and surface area are key concepts in geometry and calculus. While our specific problem only concerns surface area, it's useful to understand both concepts.

• Volume measures the space inside a 3D object.
• Surface area measures the total area of the outermost layer of the object.

In this exercise, even though we don't calculate the volume, we work extensively with surface areas. By distinguishing between inner and outer surface areas and employing them in our calculations, we find how much wood is needed. This problem shows how closely linked volume and surface area can be in determining material requirements for constructing objects.