Two of the three dimensions of a cuboidal box are 9 units and 12 units. Compute the third dimension of the box, such that the body diagonal of the box is the least possible integer under the given conditions.
A.8

B.15
C.12

D.14

To solve this problem, we need to compute the third dimension of a cuboidal box such that the body diagonal is the least possible integer. 

Given that the dimensions of the box are 9 units, 12 units, and the unknown dimension \( x \), we can use the formula for the body diagonal \( d \) of a cuboid:

\[
d = \sqrt{a^2 + b^2 + c^2}
\]

where \( a \), \( b \), and \( c \) are the three dimensions of the cuboid. In this case, \( a = 9 \), \( b = 12 \), and \( c = x \). We want the body diagonal \( d \) to be the least possible integer.

1. Plug in the known values:

\[
d = \sqrt{9^2 + 12^2 + x^2}
\]
\[
d = \sqrt{81 + 144 + x^2}
\]
\[
d = \sqrt{225 + x^2}
\]

2. We want \( \sqrt{225 + x^2} \) to be an integer. Let's try different values of \( x \) from the options provided:

- For \( x = 8 \):
  \[
  d = \sqrt{225 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17
  \]

- For \( x = 15 \):
  \[
  d = \sqrt{225 + 15^2} = \sqrt{225 + 225} = \sqrt{450}
  \]
  \( \sqrt{450} \) is not an integer.

- For \( x = 12 \):
  \[
  d = \sqrt{225 + 12^2} = \sqrt{225 + 144} = \sqrt{369}
  \]
  \( \sqrt{369} \) is not an integer.

- For \( x = 14 \):
  \[
  d = \sqrt{225 + 14^2} = \sqrt{225 + 196} = \sqrt{421}
  \]
  \( \sqrt{421} \) is not an integer.

Among the given options, \( x = 8 \) gives us the least possible integer value for the body diagonal, which is 17.

Thus, the third dimension is \( \boxed{8} \). 

Would you like further details or have any questions?

Here are 5 related questions:
1. How would you calculate the volume of a cuboid with given dimensions?
2. What is the formula for the surface area of a cuboid?
3. Can the body diagonal formula be used for any three-dimensional shape?
4. How does changing one dimension of a cuboid affect its body diagonal?
5. If two dimensions are equal, what type of special cuboid is formed?

**Tip:** The Pythagorean theorem is extended into three dimensions for calculating the body diagonal of a cuboid.

Two of the three dimensions of a cuboidal box are 9 units and 12 units. Compute the third dimension of the box, such that the body diagonal of the box is the least possible integer under the given conditions. A. 8 B. 15 C. 12 D. 14

Learn how to compute the third dimension of a cuboidal box with given dimensions of 9 and 12 units, such that the body diagonal is the least possible integer. Explore the use of the Pythagorean theorem in three dimensions to find the solution. This problem is suitable for students in Grades 9-10.

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