Answer :
The equation [tex]y=-5x^2+100x[/tex] describes the relationship between time [tex]x[/tex] (in seconds) and the speed [tex]y[/tex] (in rpm) of the ceiling fan. This is a quadratic equation, and its graph is a downward-opening parabola.
Key Features of the Equation:
- Maximum Speed:
- The maximum speed occurs at the vertex of the parabola. The [tex]x[/tex]-coordinate of the vertex can be found using the formula:
[tex]x=-\frac{b}{2a}[/tex]
- Here, [tex]a=-5[/tex] and [tex]b=100[/tex]:
[tex]x=-\frac{100}{2(-5)}=\frac{100}{10} =10[/tex]
- Substituting [tex]x=10[/tex] back into the equation gives the maximum speed:
[tex]y=-5(10)^2+100(10)=-500+1000=500 rpm[/tex]
2. Initial Speed:
- At [tex]x=0[/tex] (the moment the fan is turned on):
[tex]y=-5(0)^2+100(0)=0rpm[/tex]
3. Stopping Time:
- To find when the fan stops completely, set [tex]y=0[/tex]:
[tex]-5x^2+100x=0[/tex]
- Factor the equation:
[tex]x(-5x+100)=0[/tex]
- This gives [tex]x=0[/tex] (when the fan is first turned on) and:
[tex]-5x+100=0[/tex] ---> [tex]5x=100[/tex] ---> [tex]x=20[/tex]
- Therefore, the fan completely stops after 20 seconds.
Summary:
- Maximum Speed: 500 rpm at [tex]x=10[/tex] seconds.
- Fan Stops: The fan completely stops at [tex]x=20[/tex] seconds.