asinus

$\dfrac{s}{25\ miles/h}-\dfrac{s}{40\ miles/h}=\dfrac{45}{60}h\\ \dfrac{s(40-25)\ miles/h}{1000\ (miles/h)^2}=\dfrac{45}{60}h\\ \dfrac{15s}{1000\ miles/h}=\dfrac{45}{60}h\\ s=\dfrac{45h\cdot1000\ miles/h}{60\cdot15}\\ \color{blue}s=50\ miles$

  !

Hello tomtom, hello greenplanet2050!

The left function is an upwardly open second power parabola above the x-axis,

the left function is a downwardly open second power parabola under the x-axis.
For every x value there is no real function value that is included in both functions at the same time.

The graphs do not touch each other.

  !

Hello blackpanther!

I have corrected AX on PX.

$\frac{\angle YPM}{2}=atan\ \frac{8}{18}=23.962^\circ\\ \angle RPM=47.925^\circ\\ h_{via\ PQ}=22\cdot sin\ \angle RPM=22\cdot sin\ 47.925^\circ\\ h_{via PQ}=16.32\\ A_{\triangle PQR}=\frac{1}{2}\cdot 36\cdot 16.32\\ \color{blue}A_{\triangle PQR}=293.76$

 !

 Find AE.

Hello magenta!

Cosine law:$c^2=a^2+b^2-2ab\ cos\ \gamma$

$\overline{AE}\ ^2=3^2+5^2-2\cdot 3\cdot 5\cdot cos\ 60^{\circ}\\ \overline{AE}=\sqrt{9+25-30\cdot \frac{1}{2}}=\sqrt{19}\\ \color{blue}\overline{AE}=4.359$

 !

Find the area of the square.

Hello magenta!

$1:3=x:(3-x)\\ 3x=3-x\\ 4x=3\\ x=\frac{3}{4}\\ \color{blue}A=\frac{3}{4}\cdot \frac{3}{4}=\frac{9}{16}$

  !

Find the volume of pyramid EFGH.

Hello Vxrtate!

$\Delta EFG \cdot \frac{2h}{3}=18\\ \Delta EFG =\frac{3\cdot 18}{2h}\\ V_{EFGH}=\Delta EFG\cdot \frac{1}{3}h\\ V_{EFGH}=\frac{3\cdot 18}{2h}\cdot \frac{1}{3}h\\ \color{blue}V_{EFGH}=9$

 !

Hello tomtom!

$96-44=a+b=h(tan20^\circ +tan 55^\circ )\\ h=\dfrac{52}{tan20^\circ +tan 55^\circ}=29.016\\ \overline{LM}=\dfrac{44+96}{2}=70\\ A=\overline{LM}\cdot h=70\cdot 29.016\\ \color{blue} A=2031.12$

 !

$\cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{1 + \cfrac{1}{4}}}}.\\ =\cfrac{1}{1 + \cfrac{1}{4 + \cfrac{4}{5}}}\\ =\cfrac{1}{1 + \cfrac{5}{24 }}\\ \color{blue}=\cfrac{24}{29}$

 !

$d=\frac{24km}{t}\\ e=\frac{(24-10)km}{t}=\frac{14km}{t}\\ f=\frac{(24-15)km}{t}= \frac{9km}{t}\\ s_e=\frac{14km}{t}x=24km\\ s_f=\frac{9km}{t}x=24km\cdot \frac{9}{14}\\ \color{blue}s_e-s_f=24km-24km\cdot \frac{9}{14}=8.571km$

d,e,f are the running speeds of the runners.

t is the running time of Dirk.

x is the running time of Edith.

s_e, s_f are the routes to x.

Foley will be 8.571 km behind when Edith finishes the race.

 !

$\frac{3-z}{z+1} \ge 2 + z\\ 3-z\ge 2z+2+z^2+z\\ 0= z^2+4z-1\\ z_0=-2\pm \sqrt{4+1}\\ z_0\in \{-\sqrt{5}-2,\sqrt{5}-2\}\\ z_{undetermined}=-1$

 $\color{blue}z\in \mathbb R$ | - infinite < z <= -4,236 | and | -1 < z <= 0.236 | 

 !