asinus

A train traveling at 100 km an hour takes 3 seconds to enter a tunnel (T) and an additional thirty seconds to pass completely through it. Find the length (x), in kilometers, of the train. Express your answer as a common fraction.

Hello Guest!

\(t=\dfrac{s}{v}\)

\(3sek=\dfrac{T\cdot h}{100km}\cdot\dfrac{3600sek}{h}\\ T=\dfrac{3sek\cdot 100km}{3600sek}\)

\(T=0.08\overline{3}km\)

\(30sek=\dfrac{x\cdot h}{100km}\\ x=\dfrac{30sek\cdot 100km}{h}\cdot \dfrac{h}{3600sek}\\ \)

\(x=0.8\overline{33}km\)

The length of the train is \(\frac{5}{6}\ km=0.8\overline{33}\ km.\) 

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Hallo Nita!

Willkommen im web2.0-Matheforum!

Du fragst: Seid ihr gut in Mathe?

Stelle uns Mathefragen. Dann kannst du es selbst beurteilen.

Grüße

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Triangle ABC is isosceles with AB = BC.  If AB = 20 and [ABC] = 240, then find the perimeter of triangle ABC.

Hello Guest!

\(\beta=360°-240°=120°\)

\(b^2=a^2+c^2-2ac\ cos\beta\)

\(b^2=20^2+20^2-2\cdot 20\cdot 20\cdot \ cos\ 120°\)

\(b^2=1200\)

\(b=\sqrt{1200}\)

\(b=34.641\)

\(P=a+c+b=20+20+34.641\)

\(P=74.641\)

 The perimeter of triangle ABC is 74.641

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For what real values of a is the expression (a + 3)/(a^3 - a) undefined? List your answers in increasing order separated by commas.

Hello Guest!

(a + 3)/(a^3 - a) is undefined if the term a^3 - a is zero.

\(a^3-a=a(a^2-1)=0\\ a=0,\ a^2-1=0\\ a\in\{0,1,-1\}\)

\((a + 3)/(a^3 - a) \ is\ undefined\ | a\in\{-1,0,1\}\)

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Consider 2 functions \(f(x), g(x)\)that is defined on the interval \([0, \infty)\) where
\(f(x)=x^3+x^2+x+2\)
\(g(x)=2f^{-1}(x)-1\)
Find the value of \(g^{-1}(0)+g^{-1}(3)\).

Hello Guest!

\(g(x)=\dfrac{2}{x^3+x^2+x+2}-1\)

\(g^{-1}(x)=\dfrac{1}{\dfrac{2}{x^3+x^2+x+2}-1}\)

\(g^{-1}(0)=\dfrac{1}{\dfrac{2}{0^3+0^2+0+2}-1}=\frac{1}{0}=\infty\)

\(g^{-1}(3)=\dfrac{1}{\dfrac{2}{3^3+3^2+3+2}-1}=-1.\overline{051282}\)

\(g^{-1}(0)+g^{-1}(3)=\infty\)

2 snowplows can clear a road 12km x 4 m in 3 hours. How long will take for 10 snowplows to clear a road 2km x 12m.

 Hello Guest!

\(N=W/t\)

\(N_{sp}=\dfrac{12km\cdot 4m}{2\cdot3h}\cdot \dfrac{1000m}{km}=8000m^2/h\)

\(t=\dfrac{W}{N_{sp}}\)

\(t=\dfrac{2km\cdot 12m}{10\cdot 8000m^2/h}\cdot \dfrac{1000m}{km} \)

\(t=0.3h\cdot\dfrac{60min}{h}=\color{blue}\ 18min\)

10 snowplows clear a road of 2km x 12m in 18min.

You're right.

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For his art project, Julian wanted to make a pixel art by completely covering the 2-feet-by-3-feet poster board with colored stickers that measure 1.5-inches-by-1.5-inches without any overlap.How many stickers would Julian need?

Hello Potatobird!

\(2ft\cdot 3ft\cdot \frac{12^2\cdot inch^2}{ ft^2}:(1.5^2\cdot inch^2)=\color{blue}384\)

Julian needs 384 stickers.

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Danke Probolobo!

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The volume of the apple juice in a glass whose shape is an inverted frustum of a cone is 1176π mm³. The depth of the juice is 18mm.

Find the radius of the lower base if the product of its radii is 60 mm². *

a.5 mm

b.6 mm

c.7 mm

d.8mm

Hello Guest!

\(V=\dfrac{1}{3}\pi h(R^2+Rr+r^2)\\ 1176mm^3=\dfrac{1}{3}\cdot 18mm\cdot (R^2+60mm^2+r^2)\\\)

\(Rr=60mm^2\\ R= \dfrac{60mm^2}{r}\)

\(1176mm^3=\dfrac{1}{3}\cdot 18mm\cdot ( (\frac{60mm^2}{r})^2+60mm^2+r^2)\)

\((\frac{60mm^2}{r})^2+60mm^2+r^2=196mm^2\\ 3600mm^4+60r^2mm^2+r^4=196r^2mm^2\\ r^2=x\)

\(3600mm^4+60x\ mm^2+x^2=196x\ mm^2\\ 3600mm^4-136x\ mm^2+x^2=0\)

\(x=68\pm \sqrt{4624-3600}\\ x=68\pm 32\\ r=\sqrt{x}\\ r=\sqrt{68-32}\)

\(r=6\ mm\)

The radius of the base of the frustum of a cone is 6mm.

b. it's right.

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In triangle ABC show above, cos B = 6/10. What is tan C?

Hello Guest!

\(c^2=a^2+b^2-2ab\cdot cos\ C\\ cos\ C=\dfrac{a^2+b^2-c^2}{2ab}\\ cos\ C=\dfrac{49+32-25}{56\cdot \sqrt{2}}=\dfrac{1}{\sqrt{2}}\)

\(C=\arccos\ \dfrac{1}{\sqrt{2}}\)

\(C=45°\)

\(tan\ C=tan 45°\)

\(tan\ C=1\)

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