C. 2.5 ft
To find the width of the border, we need to subtract the area of the stained glass window from the total area of the window including the border.
Total area = (2+2x)(4+2x) = 8 + 12x + 4x + 4x^2 = 8 + 20x + 4x^2
Area of stained glass window = 2*4 = 8
Therefore, the area of the border = total area - area of stained glass window
7 = 20x + 4x^2
Rearranging the equation,
4x^2 + 20x - 7 = 0
We can solve this quadratic equation to find x. The positive value of x will be the width of the border.
The width of the border turns out to be 2.5 ft.
A rectangular stained glass window is 2 feet by 4 feet. A clear glass border is constructed around the stained glass window. The width of the border is equal and was made out of 7 square feet of clear glass. What is the width of the border?
A. 0.5 ft
B. 1.5 ft
C. 2.5 ft
D. 3.5 ft
7 answers
A rectangular pool is 20 feet wide and 50 feet long. A deck used for sunning surrounds the pool. The deck is the same width all the way around the pool. The total area of the deck is 456 square feet. How wide is the deck around the pool?
A. 2
B. 3
C. 4
D. 5
A. 2
B. 3
C. 4
D. 5
C. 4
Let the width of the deck be x feet.
The total area of the pool including the deck can be calculated as follows:
Total area = (20 + 2x)(50 + 2x)
Area of just the pool = 20 * 50 = 1000 square feet
Therefore, the area of the deck = Total area - Area of just the pool = 456
Solving (20 + 2x)(50 + 2x) - 1000 = 456
Expanding and simplifying:
1000 + 40x + 100x + 4x^2 - 1000 = 456
4x^2 + 140x = 456
4x^2 + 140x - 456 = 0
Solving the above quadratic equation:
x = 4
Therefore, the width of the deck around the pool is 4 feet.
Let the width of the deck be x feet.
The total area of the pool including the deck can be calculated as follows:
Total area = (20 + 2x)(50 + 2x)
Area of just the pool = 20 * 50 = 1000 square feet
Therefore, the area of the deck = Total area - Area of just the pool = 456
Solving (20 + 2x)(50 + 2x) - 1000 = 456
Expanding and simplifying:
1000 + 40x + 100x + 4x^2 - 1000 = 456
4x^2 + 140x = 456
4x^2 + 140x - 456 = 0
Solving the above quadratic equation:
x = 4
Therefore, the width of the deck around the pool is 4 feet.
its 3 feet
Apologies for the mistake in the initial calculation. Let's re-calculate using the correct values:
Let the width of the deck be x feet.
Total area of the deck = (20 + 2x)(50 + 2x) - 20 * 50
Given that the total area of the deck is 456 square feet:
(20 + 2x)(50 + 2x) - 1000 = 456
(1000 + 40x + 100x + 4x^2) - 1000 = 456
4x^2 + 140x = 456
Subtracting 456 from both sides:
4x^2 + 140x - 456 = 0
Now, we'll solve the quadratic equation:
4x^2 + 140x - 456 = 0
x^2 + 35x - 114 = 0
(x + 38)(x - 3) = 0
Therefore, x = 3 or x = -38.
Since x = -38 is not a valid solution for the width of the deck, the correct width of the deck around the pool is 3 feet. Thank you for pointing out the error.
Let the width of the deck be x feet.
Total area of the deck = (20 + 2x)(50 + 2x) - 20 * 50
Given that the total area of the deck is 456 square feet:
(20 + 2x)(50 + 2x) - 1000 = 456
(1000 + 40x + 100x + 4x^2) - 1000 = 456
4x^2 + 140x = 456
Subtracting 456 from both sides:
4x^2 + 140x - 456 = 0
Now, we'll solve the quadratic equation:
4x^2 + 140x - 456 = 0
x^2 + 35x - 114 = 0
(x + 38)(x - 3) = 0
Therefore, x = 3 or x = -38.
Since x = -38 is not a valid solution for the width of the deck, the correct width of the deck around the pool is 3 feet. Thank you for pointing out the error.
Consider the polynomial p(x) = 16x^4 - 81y^4. Which of the following are equivalent expressions? Select all that apply.
A. (4x^2-9y^2)(4x^2 - 9y^2)
B. (4x^2-9y^2)(4x^2 + 9y^2)
C. (4x^2+9y^2)(2x-3y)(2x-3y)
D. (4x^2+9y^2)(2x+3y)(2x+3y)
E. (4x^2+9y^2)(2x+3y)(2x-3y)
A. (4x^2-9y^2)(4x^2 - 9y^2)
B. (4x^2-9y^2)(4x^2 + 9y^2)
C. (4x^2+9y^2)(2x-3y)(2x-3y)
D. (4x^2+9y^2)(2x+3y)(2x+3y)
E. (4x^2+9y^2)(2x+3y)(2x-3y)
To factor the polynomial \( 16x^4 - 81y^4 \), we can use the difference of squares formula: \( a^2 - b^2 = (a + b)(a - b) \).
Given:
\( 16x^4 - 81y^4 = (4x^2)^2 - (9y^2)^2 = (4x^2 + 9y^2)(4x^2 - 9y^2) \)
Now let's see which of the given expressions are equivalent to \( (4x^2 + 9y^2)(4x^2 - 9y^2) \):
A. (4x^2 - 9y^2)(4x^2 - 9y^2) = Incorrect - The correct factorization is using the difference of squares formula as stated above.
B. (4x^2 - 9y^2)(4x^2 + 9y^2) = Correct - This is equivalent to the factorization \( 16x^4 - 81y^4 \).
C. (4x^2 + 9y^2)(2x - 3y)(2x - 3y) = Incorrect - This is not equivalent to the factorization of \( 16x^4 - 81y^4 \).
D. (4x^2 + 9y^2)(2x + 3y)(2x + 3y) = Incorrect - This is not equivalent to the factorization of \( 16x^4 - 81y^4 \).
E. (4x^2 + 9y^2)(2x + 3y)(2x - 3y) = Correct - This is also equivalent to the factorization \( 16x^4 - 81y^4 \).
Therefore, the equivalent expressions are B. (4x^2 - 9y^2)(4x^2 + 9y^2) and E. (4x^2 + 9y^2)(2x + 3y)(2x - 3y).
Given:
\( 16x^4 - 81y^4 = (4x^2)^2 - (9y^2)^2 = (4x^2 + 9y^2)(4x^2 - 9y^2) \)
Now let's see which of the given expressions are equivalent to \( (4x^2 + 9y^2)(4x^2 - 9y^2) \):
A. (4x^2 - 9y^2)(4x^2 - 9y^2) = Incorrect - The correct factorization is using the difference of squares formula as stated above.
B. (4x^2 - 9y^2)(4x^2 + 9y^2) = Correct - This is equivalent to the factorization \( 16x^4 - 81y^4 \).
C. (4x^2 + 9y^2)(2x - 3y)(2x - 3y) = Incorrect - This is not equivalent to the factorization of \( 16x^4 - 81y^4 \).
D. (4x^2 + 9y^2)(2x + 3y)(2x + 3y) = Incorrect - This is not equivalent to the factorization of \( 16x^4 - 81y^4 \).
E. (4x^2 + 9y^2)(2x + 3y)(2x - 3y) = Correct - This is also equivalent to the factorization \( 16x^4 - 81y^4 \).
Therefore, the equivalent expressions are B. (4x^2 - 9y^2)(4x^2 + 9y^2) and E. (4x^2 + 9y^2)(2x + 3y)(2x - 3y).