4. A roof truss is made in the shape of an inverted $\mathrm{V}$. The lengths of the two edges are 31 feet and 48 feet. The edges meet at the peak making a 63 degree angle. To the nearest foot, find the width of the truss and the height of the peak.

Step 1 ：Given that the lengths of the two edges of the truss are 31 feet and 48 feet, and they meet at a 63 degree angle at the peak.

Step 2 ：We can use the law of cosines to find the width of the truss. The law of cosines states that for any triangle with sides of lengths a, b, and c and an angle γ opposite side c, \(c^2 = a^2 + b^2 - 2ab\cos(γ)\). In this case, a and b are the lengths of the two edges of the truss and γ is the angle at the peak.

Step 3 ：Substituting the given values into the law of cosines, we find that the width of the truss, c, is approximately 44 feet.

Step 4 ：Once we have the width of the truss, we can use the Pythagorean theorem to find the height of the peak. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, one of the edges of the truss is the hypotenuse and the width of the truss and the height of the peak are the other two sides.

Step 5 ：Using the Pythagorean theorem, we find that the height of the peak, h, is approximately 22 feet.

Step 6 ：Final Answer: The width of the truss is \(\boxed{44}\) feet and the height of the peak is \(\boxed{22}\) feet.