To determine the mass of a block of metal with new dimensions, we need to take into account both the volume and mass of the original block and the new block. Here is a step-by-step solution:
1. Determine the Volume of the Initial Block:
The original block has dimensions:
- Length = 16 cm
- Width = 10 cm
- Height = 4 cm
The volume ([tex]\( V_{\text{initial}} \)[/tex]) of a rectangular block is calculated using the formula:
[tex]\[
V_{\text{initial}} = \text{Length} \times \text{Width} \times \text{Height}
\][/tex]
Substituting the given values, we get:
[tex]\[
V_{\text{initial}} = 16 \, \text{cm} \times 10 \, \text{cm} \times 4 \, \text{cm} = 640 \, \text{cm}^3
\][/tex]
2. Determine the Volume of the New Block:
The new block has dimensions:
- Length = 12 cm
- Width = 8 cm
- Height = 2 cm
The volume ([tex]\( V_{\text{new}} \)[/tex]) of the new block is calculated using the same formula:
[tex]\[
V_{\text{new}} = \text{Length} \times \text{Width} \times \text{Height}
\][/tex]
Substituting the given values, we get:
[tex]\[
V_{\text{new}} = 12 \, \text{cm} \times 8 \, \text{cm} \times 2 \, \text{cm} = 192 \, \text{cm}^3
\][/tex]
3. Find the Mass of the New Block:
The mass of the original block is given as 1760 grams.
To find the mass of the new block, we need to consider the ratio of the volume of the new block to the volume of the original block, because the mass is directly proportional to the volume for the same material.
Let [tex]\( m_{\text{new}} \)[/tex] be the mass of the new block. We use the ratio of the volumes and the mass of the original block to find [tex]\( m_{\text{new}} \)[/tex]:
[tex]\[
m_{\text{new}} = \frac{V_{\text{new}}}{V_{\text{initial}}} \times \text{mass}_{\text{initial}}
\][/tex]
Substituting the known values, we get:
[tex]\[
m_{\text{new}} = \frac{192 \, \text{cm}^3}{640 \, \text{cm}^3} \times 1760 \, \text{g}
\][/tex]
Simplify the fraction:
[tex]\[
m_{\text{new}} = \frac{3}{10} \times 1760 \, \text{g} = 0.3 \times 1760 \, \text{g}
\][/tex]
[tex]\[
m_{\text{new}} = 528 \, \text{g}
\][/tex]
Therefore, the mass of the new block of metal measuring 12 cm x 8 cm x 2 cm is 528 grams.
