It seems there was a mistake in your steps, specifically in the last part where you solve for [tex]\( w \)[/tex]. Let's correctly solve this question step-by-step.
1. Find the constant of variation [tex]\( k \)[/tex]:
- Given: weight of the first dog [tex]\( w_1 = 50 \)[/tex] pounds, medicine for the first dog [tex]\( m_1 = 0.5 \)[/tex] milligrams.
- Since the number of milligrams [tex]\( m \)[/tex] varies directly with the weight [tex]\( w \)[/tex], the relationship can be described as [tex]\( m = k \cdot w \)[/tex], where [tex]\( k \)[/tex] is the constant of variation.
- To find [tex]\( k \)[/tex]:
[tex]\[
k = \frac{m_1}{w_1} = \frac{0.5}{50} = 0.01
\][/tex]
2. Write the direct variation equation:
- Using the constant of variation [tex]\( k \)[/tex] found in the previous step, the equation becomes:
[tex]\[
m = 0.01 \cdot w
\][/tex]
3. Substitute 10 into the equation to find the dosage for a 10-pound dog:
- Given: the weight of the second dog [tex]\( w_2 = 10 \)[/tex] pounds.
- Substitute [tex]\( w_2 \)[/tex] into the equation:
[tex]\[
m = 0.01 \cdot 10
\][/tex]
4. Solve for [tex]\( m \)[/tex]:
- Perform the multiplication:
[tex]\[
m = 0.01 \cdot 10 = 0.1
\][/tex]
Therefore, the correct dosage for a 10-pound dog is [tex]\( \boxed{0.1} \)[/tex] milligrams.
