Problem 20 - Technical Mathematics with Calculus

Question

Truck Capacity. The bed of a certain pickup truck measures 58 in. by 74 in. and is 16 in. deep. How many cubic feet of sand can it hold, filled level with the sides, allowing \(1.2 \mathrm{ft}^{3}\) for each wheel well?

Step-by-Step Solution

Answer

The truck can hold the calculated volume from Step 4 minus 2.4 cubic feet for the wheel wells.

Step 1: Convert dimensions to feet

Since the given dimensions are in inches, convert them to feet by dividing by 12 (since 1 foot = 12 inches). So, the length in feet is 74 inches / 12 = 6.1667 feet, the width is 58 inches / 12 = 4.8333 feet, and the depth is 16 inches / 12 = 1.3333 feet.

Step 2: Calculate the volume of the truck bed in cubic feet

Multiply the length, width, and depth of the truck bed to find its volume in cubic feet. The volume V is given by V = length * width * depth = 6.1667 ft * 4.8333 ft * 1.3333 ft.

Step 3: Subtract the volume of the space taken by the wheel wells

Subtract the volume of space occupied by the wheel wells from the total volume of the truck bed. The total volume allowing for the wheel wells is V_total = V - 1.2 ft^3 for each wheel well.

Step 4: Calculate the final answer

Since typically there are two wheel wells, subtract 2 times 1.2 cubic feet from the total volume calculated in Step 2 to get the volume of sand the truck can hold.

Key Concepts

Unit Conversion

Unit conversion is a fundamental process in applied mathematics and various scientific disciplines. It involves changing a measurement from one unit to another without altering the actual quantity. This concept is crucial when working with measurements in real-world scenarios, such as calculating the volume of a truck bed.

For instance, to convert inches to feet in the given exercise, you divide the number in inches by 12 since there are 12 inches in a foot. Achieving fluency in these conversions is essential both in academic settings and practical applications like carpentry, construction, and even shopping for the right fit of materials.

Volume Calculation

Volume calculation is key in determining the capacity of three-dimensional spaces. It's used extensively in construction, manufacturing, and logistics to maximize efficiency and resource management.

The exercise provided demonstrates a practical volume calculation by multiplying the length, width, and depth of the truck bed. Knowing how to calculate volume is indispensable for students aiming to engage in fields like engineering or architecture and for everyday tasks such as packing a moving box or filling a pool with water.

Cubic Feet Measurement

Cubic feet measurement is a standard unit for measuring volume in the imperial system, often used in the United States. A cubic foot is the volume of a cube with sides that are exactly one foot in length. This measurement is particularly used in shipping, refrigeration, and agricultural storage.

In our exercise, after finding the dimensions in feet, we calculate the volume in cubic feet to understand how much sand the truck bed can hold. This step is essential for practical applications like determining the appropriate amount of materials needed for construction projects or landscape designs.

Applied Mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In this context, mathematics is used to solve real-world problems, like calculating the capacity of a truck bed.

Understanding the principles of applied mathematics not only aids in academic success but also empowers students to address everyday challenges. By applying mathematical concepts to a tangible scenario, such as the truck bed volume exercise, students can more easily grasp the relevance and importance of mathematics in daily life.