Tasha is planning an expansion of a square flower garden in a city park. If each side of the original garden is increased by 7 m, the new total area of the garden will be 144 m². Find the length of each side of the original garden.
A. 19 m
B. 12 m
C. 5 m
D. radical 5 m
AAAaannndd the bot gets it wrong yet again!
(x+7)^2 = 144
x+7 = 12
x = 5
I apologize for the error in my previous response. Yes, you are correct that the solution can be obtained by setting up the equation (x+7)^2 = 144 and solving for x, which gives x = 5. Therefore, the length of each side of the original garden is 5 meters.
The correct answer is C. 5 m. Thank you for bringing this to my attention.
To solve this problem, we need to set up an equation using the information given.
Let's assume that the length of each side of the original garden is "x" meters.
Since the length of each side is increased by 7 meters, the new length of each side would be "x + 7" meters.
The total area of a square is given by the equation A = side length squared.
So, the total area of the original garden would be x^2 square meters, and the total area of the expanded garden would be (x + 7)^2 square meters.
According to the problem, the new total area of the garden is 144 m². So, we can set up the equation:
(x + 7)^2 = 144
To find the value of x, we need to solve this equation.
Expanding the equation, we get:
x^2 + 14x + 49 = 144
Rearranging the equation, we have:
x^2 + 14x + 49 - 144 = 0
Combining like terms, we get:
x^2 + 14x - 95 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, it is easier to use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1 (the coefficient of x^2), b = 14, and c = -95.
Plugging these values into the quadratic formula, we get:
x = (-14 ± √(14^2 - 4(1)(-95))) / (2(1))
Simplifying further, we have:
x = (-14 ± √(196 + 380)) / 2
x = (-14 ± √576) / 2
x = (-14 ± 24) / 2
Now, we have two possible values for x:
x = (-14 + 24) / 2 = 10 / 2 = 5
and
x = (-14 - 24) / 2 = -38 / 2 = -19
Since a length cannot be negative, we discard the negative value.
Therefore, the length of each side of the original garden is 5 meters.
So, the answer is choice C. 5 m.
Let x be the length of each side of the original garden. Then the area of the original garden is x^2.
If each side is increased by 7 m, then the new length of each side is x+7. The new total area is (x+7)^2 = x^2 + 14x + 49.
We know that the new total area is 144 m², so we can set up an equation:
x^2 + 14x + 49 = 144
Simplifying this equation:
x^2 + 14x - 95 = 0
We can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in a=1, b=14, and c=-95, we get:
x = (-14 ± sqrt(14^2 - 4(1)(-95))) / 2(1)
x = (-14 ± sqrt(676)) / 2
x = (-14 ± 26) / 2
x = 6 or x = -20
Since we're looking for a length, we can ignore the negative solution and conclude that the length of each side of the original garden is 6 meters.
Answer: none of the given options (missing option E: 6 m).