## Geometry: Common Core (15th Edition)

The area of the rectangle is $48$ square centimeters.
To find the area of the rectangle, we need to know the dimensions of the rectangle and plug them into the following formula to calculate area: $A = lw$, where $A$ is the area of the rectangle, $l$ is the length of the rectangle, and $w$ is the width of the rectangle. Let us set up some expressions for this problem. Let $w = width$. If the length $l$ is seven more than three times the width, then we can express $l$ as follows: $l = 3w + 7$ We are also given that the perimeter is $38$ cm. Let us use the formula to calculate the perimeter of a rectangle to figure out its dimensions. The perimeter of a rectangle is given by the following formula: $P = 2l + 2w$, where $P$ is the perimeter, $l$ is the length, and $w$ is the width of the rectangle. Let's plug in what we know into this formula: $38 = 2(3w + 7) + 2w$ Let's use the distributive property first: $38 = 6w + 14 + 2w$ Combine like terms on the right side of the equation: $38 = 8w + 14$ Subtract $14$ from each side of the equation to isolate constants on one side of the equation: $8w = 24$ Divide each side by $8$ to solve for $w$: $w = 3$ Now that we have the value for $w$, we can substitute it into the expression for $l$ to find the length of the rectangle: $l = 3w + 7$ Substitute $3$ for $w$: $l = 3(3) + 7$ Multiply first, according to order of operations: $l = 9 + 7$ Add to solve for $l$: $l = 16$ We now have the dimensions of the rectangle, so we can substitute these values into the formula to find the area of the rectangle: $A = lw$ Substitute the values for $l$ and $w$ in the formula: $A = 3(16)$ Multiply to solve for $A$: $A = 48$ The area of the rectangle is $48$ square centimeters.