Problem 69 - Intermediate Algebra

Question

An archaeologist wants to ship a 34 -inch femur bone. Will it fit in a 4 -inch-tall box that has a square base with sides 24 inches long? (See Exercise 68 .) Verify your result.

Step-by-Step Solution

Answer

Yes, the 34-inch femur bone will fit in the box.

Step 1: Understand the Box Dimensions

The given box has a height of 4 inches, and the base is a square with each side measuring 24 inches. Therefore, the internal dimensions of the box are 24 inches long, 24 inches wide, and 4 inches tall.

Step 2: Determine the Diagonal of the Base

To determine the longest object that can be placed inside the base, calculate the diagonal of the square base. Use the Pythagorean Theorem for the diagonal of a square:\[ d = \sqrt{24^2 + 24^2} = \sqrt{2 \times (24^2)} = 24\sqrt{2} \text{ inches} \].

Step 3: Calculate the Max Length Inside the Box

To find the maximum length of an object that can fit inside the box diagonally through the 3D space, calculate the box's full diagonal. Use the formula for the diagonal of a rectangular prism:\[ L = \sqrt{l^2 + w^2 + h^2} = \sqrt{24^2 + 24^2 + 4^2} \].

Step 4: Solve for the Box's Internal Diagonal

Calculate the expression in the previous step:\[ L = \sqrt{576 + 576 + 16} = \sqrt{1168} \approx 34.18 \text{ inches} \].

Step 5: Compare Femur Length to Box Diagonal

The internal diagonal of the box is approximately 34.18 inches. Since the femur is 34 inches long, it can fit within the box as it is shorter than the diagonal of the box.

Key Concepts

Box Dimensions

When determining whether an object can fit in a box, understanding box dimensions is crucial. In this scenario, we have a box that is 4 inches tall with a base that forms a square, each side of which measures 24 inches. Thus, the internal dimensions are constrained by this base and height:

• Length: 24 inches
• Width: 24 inches
• Height: 4 inches

Knowing these basic dimensions helps guide what can initially fit inside and is a starting point for further calculations.

Diagonal of a Square

To understand the limitations of the box's base, we calculate the diagonal. The diagonal is the longest straight line that can be drawn across a square. The Pythagorean Theorem is essential here since it helps us find this value:

• For a square, each side is equal, so if each side is 24 inches, then by the theorem we solve: \[ d = \sqrt{24^2 + 24^2} = 24\sqrt{2} \text{ inches} \]

This calculation is important as it shows the maximum length an object can have while resting against one corner to the opposite within the square base.

Rectangular Prism Diagonal

The next step is to determine the full internal diagonal of the box, which is a rectangular prism. Calculating this diagonal gives us the maximum possible length of any object that could fit inside the box across its three dimensions. Again, the Pythagorean Theorem is key:

• Use the formula: \[ L = \sqrt{l^2 + w^2 + h^2} = \sqrt{24^2 + 24^2 + 4^2} \]

This calculation gives us: \[ L = \sqrt{576 + 576 + 16} = \sqrt{1168} \approx 34.18 \text{ inches} \]Therefore, the diagonal spans approximately 34.18 inches, allowing a better visualization of how a 3D object might fit.

Mathematical Problem-Solving

Solving mathematical problems effectively often demands a step-by-step approach. Breaking down complex issues into simpler parts helps ensure accuracy and clarity. In this exercise:

• We began by understanding given dimensions.
• Next, applied essential formulas for each dimension step-by-step.
• Finally, we compared our findings to the object's size.

This method assures that each necessary calculation is clear and verifiable. Here, by understanding the problem in incremental steps, the solution becomes straightforward and the conclusion is easy to verify. This is particularly beneficial in geometry problems like fitting objects into constrained spaces.