To find the length of side \( a \) in triangle \( \Delta ABC \) using the Law of Cosines, we can apply the formula:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \]
In this case, we are given:
- \( b = 7 \) inches
- \( c = 10 \) inches
- \( A = 72^\circ \)
First, we need to calculate \( \cos(72^\circ) \). You may use a calculator for this:
\[ \cos(72^\circ) \approx 0.3090 \]
Now, substitute the values into the formula:
\[ a^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(72^\circ) \]
Calculating the squares first:
\[ 7^2 = 49 \]
\[ 10^2 = 100 \]
Now plug in these values:
\[ a^2 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot 0.3090 \]
Calculating \( 2 \cdot 7 \cdot 10 \):
\[ 2 \cdot 7 \cdot 10 = 140 \]
Now, we multiply this by \( \cos(72^\circ) \):
\[ 140 \cdot 0.3090 \approx 43.26 \]
Now substitute this into the equation for \( a^2 \):
\[ a^2 = 49 + 100 - 43.26 \]
Now calculate:
\[ a^2 = 149 - 43.26 \approx 105.74 \]
Now, find \( a \) by taking the square root:
\[ a \approx \sqrt{105.74} \approx 10.3 \text{ inches} \]
Thus, the length of side \( a \) to the nearest tenth of an inch is:
\[ \boxed{10.3} \text{ in.} \]