Giselle has a block of wood that measures 14 cm by 12 cm by 10 cm. She is making a wooden ball in tech class.

a) What percent of the wood is wasted?
b) What assumptions are you making?

To answer the questions, let's first calculate the volume of the block of wood.

The volume of a rectangular block is calculated by multiplying its length, width, and height. In this case, the dimensions are given as follows:
Length = 14 cm
Width = 12 cm
Height = 10 cm

The volume (V) of the block of wood can be calculated as follows:
V = Length x Width x Height
= 14 cm x 12 cm x 10 cm
= 1,680 cm³

Now, we can move on to answering the questions.

a) To find the percentage of wood wasted, we need to determine the volume wasted. Since Giselle is making a wooden ball, we can assume that she removes a cylindrical section from the block. The ball is created by removing the maximum possible cylinder from the rectangular block.

Assuming that the cylinder is centered within the block, its height would be equal to the block's height (10 cm). The radius of the cylinder would be half of the smallest dimension of the block, which is 6 cm (half of 12 cm).

The volume (V_cylinder) of the cylinder is calculated as follows:
V_cylinder = π x r² x h
= π x (6 cm)² x 10 cm
≈ 1130.97 cm³ (rounded to two decimal places)

The volume wasted would be the difference between the block's volume and the cylinder's volume:
Volume wasted = V_block - V_cylinder
= 1,680 cm³ - 1130.97 cm³
≈ 549.03 cm³ (rounded to two decimal places)

To find the percentage of wood wasted, we divide the volume wasted by the volume of the block and multiply by 100:
Percentage of wood wasted = (Volume wasted / V_block) x 100
= (549.03 cm³ / 1,680 cm³) x 100
≈ 32.68% (rounded to two decimal places)

Therefore, approximately 32.68% of the wood is wasted.

b) The assumptions made in this calculation are:
1. The wooden ball is created by removing a cylindrical section from the rectangular block.
2. The maximum possible cylinder that can fit within the block is used.
3. The cylinder is centered within the block.
4. The removed section of the block is considered as the waste.

These assumptions may not be true in real-life scenarios and may vary depending on how the wooden ball is actually made.

Assuming the ball's diameter is the smallest dimension of the block, it is 10cm in diameter.

So, the fraction of the block occupied by the ball is

(pi/6 * 10^3)/(14*12*10) = 31%