Begin with a circular piece of paper with a4-in. radius as shows in (a).

sodni3 2020-12-27 Answered

Begin with a circular piece of paper with a4-in. radius as shows in (a). cut out a sector with an arc length of x.Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in (b).
image
image
a) Explain why the circumference of the bas eofthe cone is \(\displaystyle{8}{\left(\pi\right)}-{x}\).
b) Express the radius r as a function of x.
c) Express the height h as a function of x.
d) Express the volume V of the cone as afunction of x.

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

2k1enyvp
Answered 2020-12-28 Author has 26265 answers
a) The circumference of a circle with radius 4 is \(\displaystyle{8}\pi\). Since the circumference of the base of the cone is equal to the arc length of the remainder of the circle, the conference is \(\displaystyle{8}\pi-{x}\)
b) since r is the radius of the base of the cone, the circuference of that base is equal to \(\displaystyle{2}\pi\cdot{r}\).
From answer a) that circuference is also equal to \(\displaystyle{8}\pi-{x}\) so:
\(\displaystyle{8}\pi-{x}={2}\pi{r}\)
\(\displaystyle{r}={\frac{{{8}\pi-{x}}}{{{2}\pi}}}={4}-{\frac{{{x}}}{{{2}\pi}}}\)
c) since h, r and 4 make up a right triangle, we can use the pythagorean theorem:
\(\displaystyle{r}^{{2}}+{h}^{{2}}={16}\)
next we substitute the answer from b) for r and solve for h:
\(\displaystyle{h}=\sqrt{{{16}-{\left({4}-{\frac{{{x}}}{{{2}\pi}}}\right)}^{{2}}}}\)
d) the volume of a cone is:
\(\displaystyle{V}={\frac{{{1}}}{{{3}}}}\pi{r}^{{2}}{h}\)
which means:
\(\displaystyle{V}={\frac{{\pi}}{{{3}}}}{\left({4}-{\frac{{{x}}}{{{2}\pi}}}\right)}^{{2}}\sqrt{{{16}-{\left({4}-{\frac{{{x}}}{{{2}\pi}}}\right)}^{{2}}}}\)
Not exactly what you’re looking for?
Ask My Question
47
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2020-12-27

Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: \(x = h + r \cos(?), y = k + r \sin(?)\) Use your result to find a set of parametric equations for the line or conic section. \((When\ 0 \leq ? \leq 2?.)\) Circle: center: (6, 3), radius: 7

asked 2020-12-24

For Exercise,

a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola.

b. Graph the curve. c. Identify key features of the graph. That is. If the equation represents a circle, identify the center and radius.

If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity.

If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity.

If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. \(x^2\ +\ y^2\ −\ 4x\ −\ 6y\ +\ 1 = 0\)

asked 2021-08-13

The front (and back) of a greenhouse have the same shape and dimensions shown below. The greenhouse is 40 feet long and the angle at the top of the roof is \(\displaystyle{90}^{{\circ}}\). Determine the volume of the greenhouse in cubic feet. Explain your solution.
image

asked 2021-01-25

Find the equation of the graph for each conic in standard form. Identify the conic, the center, the co-vertex, the focus (foci), major axis, minor axis, \(a^{2}, b^{2},\ and\ c^{2}.\) For hyperbola, find the asymptotes \(9x^{2}\ -\ 4y^{2}\ +\ 54x\ +\ 32y\ +\ 119 = 0\)

asked 2021-08-15

Solve the system and graph the curves
\(\begin{cases}x^{2}+3x-y+2=0 \\ y-5x=1 \end{cases}\)

asked 2021-08-11

Find the vertices and foci of the conic section. \(\frac{x^2}{4 }− \frac{y^2}{9} = 36\)
image

asked 2021-08-14

Identify each conic using eccentricity.
(a) \(r=\frac{4}{1+3\sin \theta}\)
(b) \(r=\frac{7}{1-3\cos \theta}\)
(c) \(r=\frac{8}{6+5\cos \theta}\)
(d) \(r=\frac{3}{2-3\sin \theta}\)
image

...