Final answer:
To find the outer dimensions of the frame, subtract the inner dimensions from the outer dimensions. Set up an equation using the total area of the frame and photo. Solve the system of equations to find the values of 'l' and 'w'.
Step-by-step explanation:
To find the outer dimensions of the frame, we need to subtract the inner dimensions of the photo from the outer dimensions. Let's assume the width of the frame is 'w' inches. Since the frame is the same width all the way around, the outer length and width of the frame will be the inner length and width of the photo + 2w inches. Therefore, the outer dimensions of the frame will be (inner length + 2w) inches by (inner width + 2w) inches.
Given that the total area of the frame and photo is 315 square inches, we can set up the following equation: (inner length + 2w) * (inner width + 2w) = 315.
Since we have two variables in the equation, we cannot solve it directly. However, we can use the information given to solve for one variable in terms of the other and then substitute it back into the equation to solve. Let's assume the inner length of the photo is 'l' inches. Therefore, the inner width of the photo will also be 'l' inches.
Substituting these values into the equation, we get (l + 2w) * (l + 2w) = 315. Expanding and simplifying the equation, we get l^2 + 4lw + 4w^2 = 315.
Since we know the total area of the photo is 315 square inches, we can set up a second equation: l * l = 315. Simplifying this equation, we get l^2 = 315.
Now we have a system of two equations with two variables. We can solve this system to find the values of 'l' and 'w'.
We can start by substituting l^2 from the second equation into the first equation: 315 + 4lw + 4w^2 = 315. Simplifying this equation, we get 4lw + 4w^2 = 0. Now we can divide through by 4 to simplify further: lw + w^2 = 0. This equation suggests that either 'l' or 'w' must be equal to 0, which is not possible since both the length and width of the photo and frame have positive measurements. Therefore, this equation has no valid solutions.
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