Alright, let's solve this step-by-step.
1. Identify the unknown variables:
- Let the breadth of the cuboid be [tex]\( b \)[/tex].
- Since the length is twice the breadth, the length [tex]\( l \)[/tex] will be [tex]\( 2b \)[/tex].
- The length is also three times the height, so the height [tex]\( h \)[/tex] will be [tex]\( \frac{b}{3} \)[/tex].
2. Write the volume formula for a cuboid:
- The volume [tex]\( V \)[/tex] of a cuboid is given by the product of its length, breadth, and height.
- Therefore, [tex]\( V = l \times b \times h \)[/tex].
3. Substitute the given values:
- We know that the volume [tex]\( V \)[/tex] is 729 cubic units.
- Substituting the values of [tex]\( l \)[/tex] and [tex]\( h \)[/tex] into the volume formula, we get:
[tex]\[
729 = (2b) \times b \times \left(\frac{b}{3}\right)
\][/tex]
4. Simplify the equation:
- First, simplify the expression inside the parenthesis:
[tex]\[
729 = 2b^2 \times \left(\frac{b}{3}\right)
\][/tex]
- Now further simplify the equation:
[tex]\[
729 = \frac{2}{3} b^3
\][/tex]
5. Isolate [tex]\( b^3 \)[/tex] to solve for breadth [tex]\( b \)[/tex]:
- Multiply both sides of the equation by 3/2 to isolate [tex]\( b^3 \)[/tex]:
[tex]\[
b^3 = 729 \times \frac{3}{2}
\][/tex]
- Calculate the right side:
[tex]\[
b^3 = 1093.5
\][/tex]
- Now, solve for [tex]\( b \)[/tex] by taking the cube root of both sides:
[tex]\[
b = \sqrt[3]{1093.5}
\][/tex]
6. Find the value of [tex]\( b \)[/tex]:
- Using a calculator, we get:
[tex]\[
b \approx 10.302428182979986
\][/tex]
7. Calculate the length [tex]\( l \)[/tex] and the height [tex]\( h \)[/tex]:
- Length [tex]\( l = 2b \)[/tex]:
[tex]\[
l = 2 \times 10.302428182979986 \approx 20.60485636595997
\][/tex]
- Height [tex]\( h = \frac{b}{3} \)[/tex]:
[tex]\[
h = \frac{10.302428182979986}{3} \approx 3.4341427276599954
\][/tex]
8. Summary of results:
- The length of the cuboid is approximately [tex]\( 20.60485636595997 \)[/tex] units.
- The breadth of the cuboid is approximately [tex]\( 10.302428182979986 \)[/tex] units.
- The height of the cuboid is approximately [tex]\( 3.4341427276599954 \)[/tex] units.
