TheXSquaredFactor

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Questions 3
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 #1
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0

a) 

 

Since these are composite figures, it is probably best to think about each composite figure as each individual shape that it is comprised of. Let's first discern the perimeter of the trapezoid.

 

We know three of the four sides, and the fourth side is of equal length to its opposite side. The fourth side is not exposed, so we can ignore it from the calculation. Just add these lengths together to obtain the perimeter. Multiply this perimeter by two because there are two trapezoids in this diagram.

 

\(P_{\text{trpzd}}=(13+9+7)\text{mm}\\ P_{\text{trpzd}}=29\text{mm}\\ 2P_{\text{trpzd}}=58\text{mm}\)

 

The only length left to determine is the arc length of the circular arc. The circumference of the circle is \(C=2\pi r\), but we only want 44 of the 360 degrees of the circle. Add these perimeters together to find the final perimeter. 

 

\(\text{Arclength}=2\pi r*\frac{44}{360}; r=9\text{mm}\\ \text{Arclength}=\frac{99\pi}{45}\text{mm}\approx6.9115\text{mm}\)

 

Add these perimeters together to find the final perimeter:

 

\(2P_{\text{trpzd}}+\text{Arclength}=58\text{mm}+6.9115\text{mm}\\ 2P_{\text{trpzd}}+\text{Arclength}=64.9115\text{mm}\approx64.91\text{mm}\)

 

And that's that!

 

b)

The base of the equilateral triangle is 7cm. The collinear radii of the circular arc are also 7cm.  This means that 

\(P_{\triangle\text{ & radii}}=(7+7+7)\text{cm}\\ P_{\triangle\text{ & radii}}=21\text{cm}\)

 

The only thing left to do is find the arc length, just like the previous problem. There are two arcs this time. Make sure to take that into account.

 

\(\text{Arclength}=2\pi r*\frac{120}{360}; r=7\text{cm}\\ \text{Arclength}=\frac{14\pi}{3}\text{cm}\\ 2\text{Arclength}=\frac{28\pi}{3}\text{cm}\approx29.3215\text{cm}\)

 

Find the sum of these perimeters again:

 

\(P_{\triangle\text{ & radii}}+\text{Arclength}=21\text{cm}+29.3215\text{cm}\\ P_{\triangle\text{ & radii}}+\text{Arclength}=50.3215\text{cm}\approx50.32\text{cm}\)

TheXSquaredFactor 24 sept. 2018
 #1
avatar+2194 
+1

 

a)

In order to determine the vertical asymptote with a given rational function, you must set the denominator equal to zero and solve. Let's do that. 

 

\(x+c=0\) Set the denominator equal to zero and solve for x.
\(x=-c\) -c represents the expression that is the vertical asymptote. Since we know that the vertical asymptote is located at x=3, use the substitution property. 
\(3=-c\) Divide by -1 from both sides to solve for c.
\(c=-3\) Let's fill this information into the given rational function.
   

 

\(f(x)=\frac{ax+b}{x-3}\)

 

The horizontal asymptote has three conditions that you must always examine in any rational function. Every condition compares the degree of the numerator to the degree of the denominator. They are the following:

 

  • If the degree of the numerator is less than the degree of the denominator, then a horizontal asymptote occurs at y=0.
  • If the degree of the numerator is equal to the degree of the denominator, then a horizontal asymptote occurs at the ratio of the leading coefficients of the numerator and denominator.
  • If the degree of the numerator is greater than the degree of the denominator, then no horizontal asymptotes exist.

The degree of the numerator, in this case, is 1, and the degree of the denominator is also 1. According to the conditions above, we must find the "ratio of the leading coefficients of the numerator and denominator."

 

\(y=\frac{a}{1} \) The ratio of the leading coefficients are given on the left. We know that the horizontal asymptote of this particular rational function is at y=-4, so let's solve for by utilizing the Substitution Property of Equality.
\(-4=a\) We have determined the value of another variable, a
   

 

\(f(x)=\frac{-4x+b}{x-3}\)

 

Finally, we will use the last tidbit of information regarding this particular rational function, the location of the x-intercept. We only have one variable remaining, so we can just substitute this coordinate into the function and solve for b:

 

\(f(x)=\frac{-4x+b}{x-3}\) Substitute in the known coordinate of the x-intercept, (1,0).
\(f(1)=\frac{-4*1+b}{-1-3}\) Simplify the numerator and the denominator as much as possible. We know that f(1)=0, so substitute that into the function.
\(0=\frac{-4+b}{-4}\) Multiply by -4 on both sides. This ultimately cancels out the denominator.
\(0=-4+b\) Add 4 to both sides. 
\(b=4\)  
   

 

The final rational function, then, is \(f(x)=\frac{-4x+b}{x-3}\). You're done. 

 

b) 

 

This problem is almost exactly the same as the one I showcased above. Use the same techniques as I have showcased above, and you will be fine. 

TheXSquaredFactor 24 sept. 2018
 #1
avatar+2194 
+1

I encourage you to utilize the reference problems that you have been given already to tackle this problem. Below is the link to those reference problems:

 

1. https://web2.0calc.com/questions/3x-6y-x-3-y-6

2. https://web2.0calc.com/questions/please-answer-asap_5

3. https://web2.0calc.com/questions/please-answer-asap-please

 

These are just a few reference problems that you can use! 

 

Upon further review, it appears as if the answer I provided for the second question got flagged for something. Luckily for you, though, you have a myriad of problems to refer to anyway.

TheXSquaredFactor 16 sept. 2018