A rectangular box has dimensions of length=6, width=4, and height=5. the measure of the angle formed by a diagonal of the box with the base of the box is

Options

A. `27^(@)`
B. `35^(@)`
C. `40^(@)`
D. `44^(@)`

Explanation

To find the angle formed by a diagonal of a rectangular box with the base of the box, we can follow these steps:

### Step 1: Identify the dimensions of the box.
The dimensions of the box are:
- Length (l) = 6
- Width (w) = 4
- Height (h) = 5

### Step 2: Calculate the diagonal of the base.
The base of the box is a rectangle with dimensions length and width. We can use the Pythagorean theorem to find the diagonal (d) of the base.

Using the formula:
\[ d = \sqrt{l^2 + w^2} \]

Substituting the values:
\[ d = \sqrt{6^2 + 4^2} \]
\[ d = \sqrt{36 + 16} \]
\[ d = \sqrt{52} \]

### Step 3: Set up the triangle involving the diagonal of the box.
Now we have a right triangle where:
- One leg is the height of the box (h = 5).
- The other leg is the diagonal of the base (d = \(\sqrt{52}\)).
- The diagonal of the box serves as the hypotenuse.

### Step 4: Use the tangent function to find the angle.
We can use the tangent function to find the angle (θ) formed by the diagonal of the box with the base.

Using the formula:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is the height (h = 5) and the adjacent side is the diagonal of the base (d = \(\sqrt{52}\)).

Thus,
\[ \tan(\theta) = \frac{h}{d} = \frac{5}{\sqrt{52}} \]

### Step 5: Calculate θ using the inverse tangent function.
To find the angle θ, we take the inverse tangent:
\[ \theta = \tan^{-1}\left(\frac{5}{\sqrt{52}}\right) \]

### Step 6: Calculate the numerical value of θ.
Using a calculator:
\[ \theta \approx 35^\circ \]

### Final Answer:
The measure of the angle formed by the diagonal of the box with the base of the box is approximately \(35^\circ\).

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