Multiply base x height then divide by 2 if finding surface are if finding volume multiple base x height x width and divide by 3
90 points where at least two of the circles intersect.
<h3>Define circle.</h3>
A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle has rotational symmetry around the centre.
Given,
Four distinct circles are drawn in a plane.
Start with two circles; they can only come together in two places. The third circle contacts each of the previous two circles in two spots each, bringing the total number of intersections up to four with the addition of a third circle. The total number of intersections will rise by another 6 when a fourth circle intersects the first three. And the list goes on.
As a result, we get a recognizable, regular pattern: for each additional circle, there are two more intersections overall than in the circle before it.
The total number of intersections can be expressed as the sum because the maximum number of intersections of 10 circles must occur when each circle contacts every other circle in 2 places each.
2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 90.
90 points where at least two of the circles intersect.
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The 7 after the decimal is in the tenths place, so look at the one in the hundredths place. It is less than 5, so the 7 is not rounded up and stays 7. 877.7 is 877.71 rounded to the nearest tenth.
To solve this problem, you have to know these two special factorizations:
Knowing these tells us that if we want to rationalize the numerator. we want to use the top equation to our advantage. Let:
![\sqrt[3]{x+h}=x\\ \sqrt[3]{x}=y](https://tex.z-dn.net/?f=%20%5Csqrt%5B3%5D%7Bx%2Bh%7D%3Dx%5C%5C%20%5Csqrt%5B3%5D%7Bx%7D%3Dy%20)
That tells us that we have:
So, since we have one part of the special factorization, we need to multiply the top and the bottom by the other part, so:
So, we have:
![\frac{x+h-h}{h(\sqrt[3]{(x+h)^2}+\sqrt[3]{(x+h)(x)}+\sqrt[3]{x^2})}=\\ \frac{x}{\sqrt[3]{(x+h)^2}+\sqrt[3]{(x+h)(x)}+\sqrt[3]{x^2}}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bx%2Bh-h%7D%7Bh%28%5Csqrt%5B3%5D%7B%28x%2Bh%29%5E2%7D%2B%5Csqrt%5B3%5D%7B%28x%2Bh%29%28x%29%7D%2B%5Csqrt%5B3%5D%7Bx%5E2%7D%29%7D%3D%5C%5C%20%5Cfrac%7Bx%7D%7B%5Csqrt%5B3%5D%7B%28x%2Bh%29%5E2%7D%2B%5Csqrt%5B3%5D%7B%28x%2Bh%29%28x%29%7D%2B%5Csqrt%5B3%5D%7Bx%5E2%7D%7D%20)
That is our rational expression with a rationalized numerator.
Also, you could just mutiply by:
![\frac{1}{\sqrt[3]{x_h}-\sqrt[3]{x}} \text{ to get}\\ \frac{1}{h\sqrt[3]{x+h}-h\sqrt[3]{h}}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7Bx_h%7D-%5Csqrt%5B3%5D%7Bx%7D%7D%20%5Ctext%7B%20to%20get%7D%5C%5C%20%5Cfrac%7B1%7D%7Bh%5Csqrt%5B3%5D%7Bx%2Bh%7D-h%5Csqrt%5B3%5D%7Bh%7D%7D%20)
Either way, our expression is rationalized.
