The two of us ordered a pizza whose area was the same as the area of a square in which the sum of the squares of the four sides plus the sum of the squares of the two diagonals is 400. The pizza's area was
A. 50
B. 100
C. 150
D. 200
let the side of the square be x
let d be the diagonal,
d^2 = x^2+x^2 = 2x^2
d = √2 x
"the sum of the squares of the four sides plus the sum of the squares of the two diagonals is 400"
x^2+x^2+x^2+x^2 + 2(2x^2) = 400
8x^2 = 400
x^2 = 50
x = 5√2
area of square pizza is x^2
= 50
To find the area of the pizza, let's break down the given information step by step:
Step 1: Let's denote the length of a side of the square as "x".
Step 2: The sum of the squares of the four sides of the square is given as "x^2 + x^2 + x^2 + x^2".
Step 3: The sum of the squares of the diagonals of the square is given as "x^2 + x^2".
Step 4: Adding the two sums together, we have "x^2 + x^2 + x^2 + x^2 + x^2 + x^2".
Step 5: Simplifying the expression, we have "6x^2".
Step 6: According to the given information, the sum of the squares of the sides and diagonals is equal to 400.
Step 7: Setting up an equation, we have "6x^2 = 400".
Step 8: Solving for "x^2", we divide both sides of the equation by 6, giving us "x^2 = 400/6" or "x^2 = 66.67".
Step 9: Taking the square root of both sides to find the value of "x", we have "x = sqrt(66.67)".
Step 10: Since we need to find the area of the pizza, we need to find the square of the length of one side of the square.
Step 11: Calculating the square of "x", we have "x^2 = (sqrt(66.67))^2 = 66.67".
Therefore, the area of the pizza is 66.67. However, none of the choices given match this value exactly. Therefore, it seems like there might be an error in the question or the choices provided.
To find the answer to this question, we need to first understand the given information.
Let's start with the square. The sum of the squares of the four sides of a square is simply four times the square of its side length, while the sum of the squares of the two diagonals is twice the square of one side length.
Let's represent the side length of the square as "s." We can then create the equation based on the information given:
4s^2 + 2s^2 = 400
Combining like terms, we have:
6s^2 = 400
Next, we can solve for the value of "s" by dividing both sides of the equation by 6:
s^2 = 400/6
Simplifying further, we find:
s^2 ≈ 66.67
To find the area of the square, we square the side length:
Area = s^2 ≈ 66.67
Now, let's move on to the pizza. The question states that the area of the pizza is the same as the area of the square. Therefore, the area of the pizza is also approximately 66.67.
Comparing this value to the options given, we can see that the closest answer is:
A. 50
So, the correct answer is A. 50.