To find the height of the envelope, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides.
Let \( w \) be the width (7 inches) and \( h \) be the height. The diagonal \( d \) measures 10 inches. According to the Pythagorean theorem:
\[ d^2 = w^2 + h^2 \]
Substituting in the values we have:
\[ 10^2 = 7^2 + h^2 \]
Calculating the squares:
\[ 100 = 49 + h^2 \]
Now, we can isolate \( h^2 \):
\[ h^2 = 100 - 49 \] \[ h^2 = 51 \]
Next, take the square root of both sides to find \( h \):
\[ h = \sqrt{51} \]
Calculating \( \sqrt{51} \):
\[ h \approx 7.14 \]
Rounding to the nearest tenth, the height \( h \) is approximately:
\[ \boxed{7.1} \text{ inches} \]