An architect has allocated a rectangular space of 275ft^2 for a square dining room and a 14ft wide kitchen. Find both the width of the square dining room and the length of the entire rectangular space.

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Hint. If the dining room has side x, then one side of the kitchen must also be s, and the width is 14. So, just add up the areas:

x^2 + 14x = 275

To find the width of the square dining room and the length of the entire rectangular space, we can break down the problem into steps.

Let's assume the width of the square dining room is x ft. Since it is a square, the length will also be equal to x ft.

The area of the square dining room is given by multiplying the width and length, so:

Area of square dining room = x ft * x ft = x^2 ft^2

Now, the remaining rectangular space is equal to the total area minus the area of the square dining room.

Remaining rectangular space = Total area - Area of square dining room

Given that the total area is 275 ft^2, we have:

Remaining rectangular space = 275 ft^2 - x^2 ft^2

This remaining rectangular space is the area of the kitchen, which is 14 ft wide. Therefore, we can set up the following equation:

Remaining rectangular space = Length * Width
275 ft^2 - x^2 ft^2 = Length * 14 ft

Now, we can solve this equation to find the length of the entire rectangular space.

To do that, rearrange the equation:

Length = (275 ft^2 - x^2 ft^2) / 14 ft

So, the length of the entire rectangular space is:

Length = (275 - x^2) / 14 ft

Now, to find both the width of the square dining room (x ft) and the length of the entire rectangular space (Length ft), we need to solve the equation for x by substituting the expression for Length:

Length = (275 - x^2) / 14
14 * Length = 275 - x^2
x^2 = 275 - 14 * Length
x = √(275 - 14 * Length)

Now, we have an equation to calculate the width of the square dining room (x) based on the length of the entire rectangular space (Length).

To find the specific values, you can substitute different values for Length and calculate the corresponding width using the above equation.