The area of a circle is given by the expression pi(16x^2 - 40x + 25) .
a) What is the diameter of the circle?
b) What is its circumference?
area= PI (4x-5)^2
but area= PI r^2
so r= 4x-5
diameter= 2*r
circumference= 2PI*r
a) To find the diameter of the circle, we first need to find the radius.
The expression inside the parentheses is 16x^2 - 40x + 25.
The area of a circle is given by the formula A = πr^2, where r is the radius.
So, we can equate the area expression to the formula and solve for r:
π(16x^2 - 40x + 25) = πr^2
Dividing both sides by π:
16x^2 - 40x + 25 = r^2
To find the radius, we take the square root of both sides:
√(16x^2 - 40x + 25) = √r^2
Simplifying:
√(16x^2 - 40x + 25) = r
Therefore, the radius of the circle is √(16x^2 - 40x + 25).
The diameter of a circle is twice the radius. So, the diameter is:
2 * √(16x^2 - 40x + 25)
b) The circumference of a circle is given by the formula C = 2πr, where r is the radius.
To find the circumference, we can substitute the value of r we obtained earlier:
C = 2π * √(16x^2 - 40x + 25)
Therefore, the circumference of the circle is 2π * √(16x^2 - 40x + 25).
To find the diameter of a circle, you need to know the formula for the area. The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.
In this case, the expression given is pi(16x^2 - 40x + 25), which represents the area of a circle.
a) To find the diameter, we can use the formula D = 2r, where D represents the diameter and r represents the radius.
From the formula A = πr^2, we can match it to the given expression:
π(16x^2 - 40x + 25) = πr^2
We can equate the two expressions and solve for r:
16x^2 - 40x + 25 = r^2
From this equation, we can see that r^2 = 16x^2 - 40x + 25. Taking the square root of both sides gives us:
r = √(16x^2 - 40x + 25)
Now that we have the value of r, to find the diameter, we can substitute r into the formula D = 2r:
D = 2√(16x^2 - 40x + 25)
Therefore, the diameter of the circle is 2√(16x^2 - 40x + 25).
b) To find the circumference of the circle, we use the formula C = 2πr, where C represents the circumference and r represents the radius.
From the formula A = πr^2, we know that the area is equivalent to π(16x^2 - 40x + 25). We can substitute the value of r from earlier into this formula:
A = πr^2 = π(16x^2 - 40x + 25)
Now, to find the circumference, we can use the formula C = 2πr:
C = 2πr = 2π√(16x^2 - 40x + 25)
Therefore, the circumference of the circle is 2π√(16x^2 - 40x + 25).