To find the height of the gift box, we start with the information given:
- The volume of the box is \( V = 343 \) cubic inches.
- The base of the box is a square with an area of \( A = 49 \) square inches.
Since the base of the box is a square, we can denote the side length of the base as \( s \). The area of the square base is given by the formula: \[ A = s^2 \] Setting this equal to the area provided: \[ s^2 = 49 \] Taking the square root of both sides, we find: \[ s = \sqrt{49} = 7 \text{ inches} \]
Next, we use the volume formula for a rectangular prism, which is given by: \[ V = \text{Base Area} \times \text{Height} \] In this case, the base area is the area of the square base, which we already know to be \( 49 \) square inches. Let \( h \) represent the height of the box. Then we can write: \[ V = A \times h = 49 \times h \]
Setting this equal to the volume of the box, we have: \[ 49h = 343 \]
To find the height, we divide both sides by \( 49 \): \[ h = \frac{343}{49} \]
Calculating the right side, we perform the division: \[ h = 7 \]
Thus, the height of the gift box is: \[ \boxed{7} \text{ inches} \]