Answer:
65 feet
Explanation:
Let x be the length of the expansion to the north, and y be the width of the expansion to the east.
The current patio has an area of 30 * 30 = 900 square feet.
The total area of the expanded patio will be 2500 square feet.
So, we need to find values of x and y that satisfy the following two conditions:
The area of the expanded patio is 2500 square feet:
xy = 2500
The expanded patio has walls on two sides, so it can only expand to the north and east:
x + 30 = y
We can solve the second equation for x:
x = y - 30
Substitute this expression for x in the first equation:
(y - 30)y = 2500
Simplifying this equation:
y^2 - 30y - 2500 = 0
We can solve for y using the quadratic formula:
y = (30 ± sqrt(30^2 + 4*2500)) / 2
y = (30 ± sqrt(10000)) / 2
y = (30 ± 100) / 2
Since y must be positive, we can ignore the negative solution:
y = (30 + 100) / 2 = 65
Substitute this value for y in the equation x + 30 = y:
x + 30 = 65
x = 35
Therefore, the expansion to the north should be 35 feet, and the expansion to the east should be 65 feet.