A truck rental company rents a 12 ft by 8 ft by 6 ft truck for per day plus mileage. A customer prefers to rent a
less expensive smaller truck whose dimensions are smaller on each side. If the volume of the smaller truck is
, determine the dimensions of the smaller truck.

Ask by Tucker Mills. in the United States

Mar 16,2025

Question

A truck rental company rents a 12 ft by 8 ft by 6 ft truck for per day plus mileage. A customer prefers to rent a
less expensive smaller truck whose dimensions are smaller on each side. If the volume of the smaller truck is
, determine the dimensions of the smaller truck.

Ask by Tucker Mills. in the United States

Mar 16,2025

UpStudy AI · Accepted Answer

We are given that the original truck has dimensions $$12\,\text{ft}$$, $$8\,\text{ft}$$, and $$6\,\text{ft}$$ and that the smaller truck’s dimensions are reduced by $$x$$ feet on each side. Thus, the dimensions of the smaller truck are

$$
(12-x) \text{ ft},\quad (8-x) \text{ ft},\quad (6-x) \text{ ft}.
$$

The volume of the smaller truck is given as

$$
(12-x)(8-x)(6-x) = 240 \text{ ft}^3.
$$

**Step 1. Form the Volume Equation**

$$
(12-x)(8-x)(6-x) = 240.
$$

**Step 2. Expand the Equation**

First, multiply the first two factors:

$$
(12-x)(8-x) = 12\cdot8 - 12\cdot x - 8\cdot x + x^2 = 96 - 20x + x^2.
$$

Now multiply this result by the third factor $$(6-x)$$:

$$
(96 - 20x + x^2)(6-x) = 96(6-x) - 20x(6-x) + x^2(6-x).
$$

Expand each term:

$$
96(6-x) = 576 - 96x,
$$
$$
-20x(6-x) = -120x + 20x^2,
$$
$$
x^2(6-x) = 6x^2 - x^3.
$$

Combine the like terms:

$$
576 - 96x - 120x + 20x^2 + 6x^2 - x^3 = 576 - 216x + 26x^2 - x^3.
$$

So the equation becomes:

$$
576 - 216x + 26x^2 - x^3 = 240.
$$

**Step 3. Rearrange the Equation**

Subtract 240 from both sides:

$$
576 - 216x + 26x^2 - x^3 - 240 = 0,
$$

which simplifies to:

$$
- x^3 + 26x^2 - 216x + 336 = 0.
$$

Multiply the entire equation by $$-1$$ to make the leading coefficient positive:

$$
x^3 - 26x^2 + 216x - 336 = 0.
$$

**Step 4. Factor the Cubic Equation**

We look for a rational root. Testing $$x=2$$:

$$
2^3 - 26(2)^2 + 216(2) - 336 = 8 - 104 + 432 - 336 = 0.
$$

Thus, $$x = 2$$ is a root, and we can factor the cubic as:

$$
(x-2)(x^2 - 24x + 168) = 0.
$$

**Step 5. Find the Real Solution**

The quadratic factor is

$$
x^2 - 24x + 168 = 0.
$$

Calculate the discriminant:

$$
\Delta = (-24)^2 - 4(1)(168) = 576 - 672 = -96.
$$

Since the discriminant is negative, there are no additional real roots.

Thus, the only real solution is:

$$
x = 2.
$$

**Step 6. Determine the Dimensions of the Smaller Truck**

Substitute $$x = 2$$ into the dimensions:

$$
12-x = 12-2 = 10\,\text{ft},
$$
$$
8-x = 8-2 = 6\,\text{ft},
$$
$$
6-x = 6-2 = 4\,\text{ft}.
$$

Therefore, the dimensions of the smaller truck are:

$$
10\,\text{ft} \times 6\,\text{ft} \times 4\,\text{ft}.
$$
The smaller truck has dimensions of 10 ft by 6 ft by 4 ft.

A truck rental company rents a 12 ft by 8 ft by 6 ft truck for per day plus mileage. A customer prefers to rent a
less expensive smaller truck whose dimensions are smaller on each side. If the volume of the smaller truck is
, determine the dimensions of the smaller truck.

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A truck rental company rents a 12 ft by 8 ft by 6 ft truck for per day plus mileage. A customer prefers to rent a less expensive smaller truck whose dimensions are smaller on each side. If the volume of the smaller truck is , determine the dimensions of the smaller truck.

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A truck rental company rents a 12 ft by 8 ft by 6 ft truck for per day plus mileage. A customer prefers to rent a
less expensive smaller truck whose dimensions are smaller on each side. If the volume of the smaller truck is
, determine the dimensions of the smaller truck.