asinus

What us the easiest way to find the solution to 1+2-3+4+5-6+7+8-9+...+208+209-210 ?

Hello Guest!

The easiest way to do this is with the Gaussian empirical formula.

\(\sum\limits_{k=1}^{n} k =\dfrac{n(n+1)}{2}\)

\(\sum\limits_{k=1}^{209} k-210 ={\color{black}\dfrac{209(209+1)}{2}-210=}21735\)

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Rectangle ABCD is the base of pyramid PABCD. If AB=8, BC=4, PA perpendicular to AD, PA perpendicular to AB, and PB=15. Then what is the volume of PABCD?

Hello Guest!

\(h=\overline {PA} =\sqrt{15^2-8^2}\)

\(h=\sqrt{161}\)

\(V=\frac{1}{3}\cdot \overline {AB}\cdot \overline {BC}\cdot h=\frac{1}{3}\cdot 8\cdot 4\cdot \sqrt{161}\)

\({\color{blue}V=10.\overline 6\cdot\sqrt{161}}\approx135.345\)

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If the tickets were the same amount of money, how much money does Bill have?

Hello Guest!

\(\dfrac{2\cdot 20€}{5}=\dfrac{4y}{15}\\ y=\dfrac{15\cdot 2\cdot 20€}{4\cdot5}\)

\(y=30€\)

Bill has 30 Dollars.

Mr and Mrs. Wilson have 3 children: Jessica, Jeremy, and Jenna. The average of Jessica and Jeremy's age is 11. The average of all three children's ages is 10 . How old is Jenna?

Hello Guest!

\(\frac{js+jr}{2}=11\ |\ \cdot 2\\ js+jr=22\\ \dfrac{js+jr+jn}{3 }=10\ |\ \cdot 3\\ js+jr+jn=30\ | -jn\\ js+jr=30-jn\)

\(22=30-jn\\ jn=30-22\)

\(jn=8\)

Jenna is 8.

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The equation of the line passing through P2 (1,11) and P1(-4,5) can be expressed in the form x/a + y/b = 1. Find a.

Hello Guest!

Point-direction equation

\(\color{blue}y=m(x-x_1)+y_1\\ m=\dfrac{y_2-y_1}{x_2-x_1}\)

\(y=\frac{11-5}{1+4}(x+4)+5\\ y=\frac{6}{5}(x+4)+5\\ y=1.2x+4.8+5\\ 1.2x-y=-9.8\ |\ :(-9.8)\\ -\dfrac{0.6x}{4.9}+\dfrac{y}{9.8}=1\)

\(\dfrac{x}{-8.1\overline 6}+\dfrac{y}{9.8}=1\\ \\ a=-8.1\overline 6\)

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Let k be a positive real number. The line x + y = k and the circle x^2 + y^2 = 2k + 1 are drawn. Find k so that the line is tangent to the circle.

Hello Guest!

\(x^2 + y^2 = 2k + 1\\ y=\sqrt{2k+1-x^2}\\ r^2=2k+1\\ r=\sqrt{2k+1}\)

\(x+y=k\\ y=k-x\)

Distance of the straight line \(y=k-x\)  from the origin of the coordinates.

\(r=\frac{k}{\sqrt{2}}\)

\(\dfrac{k}{\sqrt{2}}=\sqrt{2k+1}\\ \dfrac{k^2}{2}=2k+1\\k^2-4k-2=0\\ k=2\pm \sqrt{4+2}\\ k=2\pm 2.449\)

\(k=4.449\)

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Michelle made some bookmarks for fundraising day. She made 1/6 of the bookmarks on the first day and 48 bookmarks on the second day. The number of bookmarks that she made on the second day was 1/5 more than the number of bookmarks she made on the first day. How many bookmarks did Michelle make for the fundraising day?

Hello RunningMan!

\(\dfrac{x}{16}(1 +\frac{1}{5})= 48\\ \dfrac{x\cdot 6}{16\cdot 5}=48\)

\(x=640\)

640 bookmarks did Michelle make for the fundraising day.

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Find all values of z such that z^4 - 4z^2 +4 = 0.

Hello Guest!

\(Replace\ z^2\ with\ x.\)

\(x^2-4x+4=0\)

       p          q

\(x=-\frac{p}{2}\pm \sqrt{(\frac{p}{2})^2-q}\\ x=2\pm \sqrt{2^2-4}\\ x=2\)

\(Replace\ x\ with\ z ^ 2. \)

\(z^2=2\\ z=\pm \sqrt{2}\)

\(z\in \{-\sqrt{2},\sqrt{2}\}\)

\(z\in \{-1.414,1.414\}\)

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Hallo Gast!

Let

f(x) = 3x + 5 if x < -3

f(x) = 7x + 4 if x >= 3

Find f(5).

Hello Guest!

5 > 3

f(5) = 7x + 4 = 7 * 5 + 4

f(5) = 39

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