what is the expression for the area of the shaded region in the diagram in factored form?
diagram:
(shaded) Large circle have diameter of 8x
(clear) small circle have diameter of 8y
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my work:
Area of large circle - Area of small circle
Pi * 4^2 - pi * 4^2 = pi ( 2x+2y)(2x-2y)
my work correct?
Yes, your work is correct. The expression for the area of the shaded region in factored form is pi(2x + 2y)(2x - 2y).
Your work is partially correct. However, there is a small mistake in the factored form of the expression for the area of the shaded region. Let's go through the correct steps to find the expression for the area.
To find the area of the shaded region, we need to subtract the area of the small circle from the area of the large circle.
The formula to find the area of a circle is A = πr^2, where A represents the area and r represents the radius.
From the information given, we know that the large circle has a diameter of 8x, so the radius would be half of that, which is 4x. Therefore, the area of the large circle would be π(4x)^2.
For the small circle, we know that its diameter is 8y, so the radius would be half of that, which is 4y. Thus, the area of the small circle would be π(4y)^2.
Now, let's subtract the area of the small circle from the area of the large circle:
Area of shaded region = Area of large circle - Area of small circle
= π(4x)^2 - π(4y)^2
= π(16x^2) - π(16y^2)
Now, let's factor out π from both terms:
Area of shaded region = π(16x^2 - 16y^2)
Therefore, the expression for the area of the shaded region in factored form is π(16x^2 - 16y^2).
pi*(4x)^2 - pi(4y)^2
= 16pi x^2 - 16pi y^2
= 16pi (x+y)(x-y)
Your mistake was in saying that
pi(4x)^2 = pi*4x^2
Bzzzzt
Parentheses are yur friends.