Question

cubic centimeters?

image of a rectangular prism with dimensions: height \\(2\sqrt{3}\text{ cm}\\), length \\(5\sqrt{3}\text{ cm}\\), and width \\(7\sqrt{5}\text{ cm}\\)

enter the correct answer in the box using simplified radical form. do not label your answer.

Explanation:

⚡ Using what you learned: simplifying radical expressions · 🆕 New Concept: Volume of a Rectangular Prism
How much space is inside a 3D box

Step 1: Identify the volume formula

The volume \( V \) of a rectangular prism is calculated by multiplying its length, width, and height:

$$ V = l \cdot w \cdot h $$

From the given image, the dimensions are:

• Length (\( l \)) = \( 5\sqrt{3}\text{ cm} \)
• Width (\( w \)) = \( 7\sqrt{5}\text{ cm} \)
• Height (\( h \)) = \( 2\sqrt{3}\text{ cm} \)

Step 2: Set up the multiplication

Substitute the dimensions into the volume formula:

$$ V = (5\sqrt{3}) \cdot (7\sqrt{5}) \cdot (2\sqrt{3}) $$

Step 3: Multiply the coefficients and the radicals

Group the whole numbers (coefficients) together and the radical parts together:

$$ V = (5 \cdot 7 \cdot 2) \cdot (\sqrt{3} \cdot \sqrt{5} \cdot \sqrt{3}) $$

Calculate the product of the coefficients:

$$ 5 \cdot 7 \cdot 2 = 70 $$

Calculate the product of the radicals:

$$ \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{3} = \sqrt{3 \cdot 5 \cdot 3} = \sqrt{45} $$

Combine them:

$$ V = 70\sqrt{45} $$

Step 4: Simplify the radical expression

To simplify \(\sqrt{45}\), find the largest perfect square factor of \(45\), which is \(9\):

$$ \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} $$

Now, multiply this back by the coefficient \(70\):

$$ V = 70 \cdot 3\sqrt{5} $$

$$ V = 210\sqrt{5} $$

Answer:

210\sqrt{5}