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8_murik_8 [283]
3 years ago
9

Help please and simplify answer in radical form if needed

Mathematics
1 answer:
deff fn [24]3 years ago
6 0

Answer:

16\sqrt{3}\:\mathrm{units^2}

Step-by-step explanation:

The area of an equilateral triangle with side length s is equal to \frac{\sqrt{3}}{4}s^2. Since the perimeter is 24 units and there are three equal sides to an equilateral triangle, each side is 24\div 3=8 units.

Substituting in this value, we get:

\frac{\sqrt{3}}{4}8^2=\frac{64\sqrt{3}}{4}=\boxed{16\sqrt{3}}

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sum of two numbers x and y is 140

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An 11-ft ladder leans against the side of a house. The bottom of the ladder is 9-ft from the side of the house. How high is the
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6.3 ft.

Step-by-step explanation:

This requires the pythagorean theorem: a^2+b^2=c^2 where "a" and "b" are legs and c is the hypotenuse(longest side of the triangle which is opposite to the 90 degree angle.) If we fill the equation where c is the ladder length(11) and "a" is the distance from the house(), and "b" is the height, 9^2+b^2=11^2 or 81+b^2=121. This is equivalent to b^2=40. b=6.3 when rounded to the nearest tenth.

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Find the partial fraction decomposition of the rational expression with prime quadratic factors in the denominator
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\dfrac{5x^4-7x^3-12x^2+6x+21}{(x-3)(x^2-2)^2}=\dfrac{a_1}{x-3}+\dfrac{a_2x+a_3}{x^2-2}+\dfrac{a_4x+a_5}{(x^2-2)^2}
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When x=\sqrt2 or x=-\sqrt2, you're left with

\begin{cases}17-8\sqrt2=(\sqrt2a_4+a_5)(\sqrt2-3)&\text{for }x=\sqrt2\\17+8\sqrt2=(-\sqrt2a_4+a_5)(-\sqrt2-3)\end{cases}\implies\begin{cases}-5+\sqrt2=\sqrt2a_4+a_5\\-5-\sqrt2=-\sqrt2a_4+a_5\end{cases}

Adding the two equations together gives -10=2a_5, or a_5=-5. Subtracting them gives 2\sqrt2=2\sqrt2a_4, a_4=1.

Now, you have

5x^4-7x^3-12x^2+6x+21=3(x^2-2)^2+(a_2x+a_3)(x-3)(x^2-2)+(x-5)(x-3)
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So the partial fraction decomposition is

\dfrac3{x-3}+\dfrac{2x-1}{x^2-2}+\dfrac{x-5}{(x^2-2)^2}
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