1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
olchik [2.2K]
2 years ago
11

A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the tr

uncated cone. What is the radius of the sphere
Mathematics
1 answer:
MariettaO [177]2 years ago
7 0
<h3>Answer:  6 units</h3>

===================================================

Explanation:

A truncated cone is where we start with a regular 3D cone and chop off the top. The portion up top is a smaller cone, which we'll ignore. The bottom part is the truncated cone portion.

In the real world, a lampshade is one example of a truncated cone.

----------------

Let x be the radius of the sphere.

We'll be focusing on a vertical cross section of the truncated cone. Refer to figure 1 in the diagram below.

We have the following points

  • A = center of the sphere
  • B = center of the circular base of the truncated cone
  • C = point 18 units to the right of point B
  • E = point directly above point A, and its the center of the circular top of the truncated cone
  • D = point 2 units to the right of point E
  • F = the location where the circle touches the slanted curved side of the truncated cone
  • G = point directly below point D, and located on segment BC

We'll connect a few of those points forming the dashed lines in figure 1.

To start off, draw a segment from D to G. This forms rectangle BGDE with sides of EB = 2x and BG = 2. The side BC is 18 units, so that must mean GC = BC - BG = 18 - 2 = 16.

Since EB = 2x, this means DG is also 2x.

------------------

Now focus on triangles ABC and AFC. They are congruent right triangles. We can prove this using the HL (hypotenuse leg) theorem. Recall that the radius of a circle is perpendicular to the tangent line, which is why angle AFC is 90 degrees. Angle ABC is a similar story.

Because they are congruent right triangles, this indicates side BC is the same length as side FC. Therefore, FC = 18

Through similar logic, triangles ADE and ADF are congruent as well which leads to ED = FD = 2.

Combine sides FD and FC to get the length of DC

DC = FD+FC = 2+18 = 20

This is the hypotenuse of the right triangle GCD

------------------

After all that, we have the right triangle GCD with the legs of 2x and 16. The hypotenuse is 20. Refer to figure 2 shown below.

As you can probably guess, we'll use the pythagorean theorem to find the value of x.

a^2 + b^2 = c^2

(DG)^2 + (GC)^2 = (CD)^2

(2x)^2 + (16)^2 = (20)^2

4x^2 + 256 = 400

4x^2 = 400-256

4x^2 = 144

x^2 = 144/4

x^2 = 36

x = sqrt(36)

x = 6 is the radius of the sphere.

You might be interested in
Can someone explain how to get the answer I’m confused on how to solve
Allisa [31]

Answer:

x=29

Step-by-step explanation:

The square notation indicates 90 degrees.

90=(x+3)+(x-1)+(x+1)

90=3x+3

87=3x

x=29

3 0
1 year ago
Read 2 more answers
What is the difference?<br> -6-(11)<br> 1-17<br> -5<br> O 5<br> 017
Roman55 [17]

Answer:

-17

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Pretty please Help me
Dahasolnce [82]
It is from 0-3 seconds
3 0
3 years ago
Read 2 more answers
Genber wiil evaluate an 8th degree polynomial in x at x=10 using the remainder theorem and synthetic division.how many coefficie
lisabon 2012 [21]
An 8th-degree polynomial needs 9 terms that involve
 x⁸, x⁷, ..., x¹, and x⁰.

x=10 implies that (x-10) is a factor of the polynomial according to the Remainder theorem.

Let the polynomial be of the form
f(x) = a₁x⁸ + a₂x⁷ + a₃x⁶ +a₄x⁵ + a₅x⁴ + a₆x³ + a₇x² + a₈x + a₉

The first few lines of the synthetic division are

10 | a₁  a₂  a₃  a₄  a₅  a₆  a₇  a₈  a₉             ( the first row has 9  coefficients)

    -----------------------------------------
      a₁

Answer:
The first row has 9  coefficients.
8 0
3 years ago
What are two ways to get from 5 to 10 ?
Aleksandr [31]
Add 5 to 5, or multiple 5 by 2.
5 0
3 years ago
Read 2 more answers
Other questions:
  • What is the pattern of the table!<br> (wait adding picture)
    11·1 answer
  • Determine the maximum number of zeros. Then, determine the x-intercepts of the function. x^3-3x^2-22x+24
    10·1 answer
  • At the zoo,there were 3times as many monkeys as lions.Tom counted a total of 24 monkeys and lions.How many monkeys were there?
    12·2 answers
  • What is 4^2+8x-11x+6-5x^2+2​
    9·1 answer
  • Identify the arc length of TV◠ in terms of π.
    7·1 answer
  • How many lines of symmetry does the pentagon in dc have??
    11·1 answer
  • Anyone plsss helpppp meeee
    14·1 answer
  • Solve the equation for c<br> M+3c=8
    14·2 answers
  • What is the Pythagorean theorm
    7·2 answers
  • The graph below shows the same shape was pushing the blue graph
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!