Find the dimensions of the right circular cylinder of greatest volume which can be inscribe in a right circular cone with radius of 5 inches and a hieght of 12 inches(hint:use similar triangle property).also draw a label diagram....plz answer its too urgent
The demand function for a product is modeled by :
p=10,000(1-(3/3+e^-0.001x)
Find the price of the product if the quantity demanded is (a) x=1000 units and (b) x=1500 units. What is the limit of the price as x increases without bound?
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Just change the numbers.
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To find the dimensions of the right circular cylinder of the greatest volume that can be inscribed in a right circular cone, we can use the property of similar triangles.
Let's start by drawing a labeled diagram to visualize the problem:
```
A
----------------
/ | \
/ |h \
/ | \
/ | \
/ | \
/ | \
B ------------- C -------------- D
```
In the diagram, we have a right circular cone where:
- Point A is the vertex of the cone.
- BC is the base of the cone with a radius of 5 inches.
- AD is the height of the cone, which is 12 inches.
Now, let's consider a right circular cylinder inscribed in the cone, such that:
- Points A, B, and D lie on the same line.
- The base of the cylinder is tangent to the base of the cone (BC).
- The height of the cylinder is equal to the height of the cone (AD).
To find the dimensions of the cylinder, let's consider the similar triangles ABD and ACD.
Using the property of similar triangles, we have:
AB / AD = AC / CD
Since AB is the radius of the cylinder (r) and AD is the height of the cone and cylinder (h), we can rewrite the equation as:
r / h = AC / CD
Substituting the given values, r = 5 inches and h = 12 inches, we can solve for AC:
5 / 12 = AC / CD
Cross-multiplying, we get:
AC = (5/12) * CD
Since the radius of the cone (BC) is 5 inches, we can write:
AC = CD - 5
Substituting this back into the equation above, we have:
(5/12) * CD = CD - 5
Multiplying both sides by 12:
5 * CD = 12 * (CD - 5)
Simplifying:
5CD = 12CD - 60
-7CD = -60
CD = 60 / 7
Therefore, CD is approximately 8.571 inches.
Since AC = CD - 5, we have:
AC = 8.571 - 5
AC ≈ 3.571 inches.
Now that we have the dimensions of the cone, we can write the dimensions of the cylinder:
Radius of the cylinder (r) = AC ≈ 3.571 inches
Height of the cylinder (h) = AD = 12 inches
Hence, the dimensions of the right circular cylinder of greatest volume that can be inscribed in the right circular cone are approximately: Radius = 3.571 inches, Height = 12 inches.
To find the dimensions of the right circular cylinder of greatest volume inscribed in a cone, we can apply the property of similar triangles.
Let's start by drawing a labeled diagram:
```
C__
/ \
/ \
A /________\ B
```
In this diagram, we have a right circular cone ABC, where AB is the base and C is the apex. The radius of the cone is 5 inches, and the height is 12 inches.
Now, we want to find the dimensions of the right circular cylinder inscribed inside the cone. Let's label the relevant points on the diagram:
- The center of the base of the cone as O.
- The center of the base of the cylinder as O'.
- The radius of the base of the cylinder as r.
- The height of the cylinder as h.
Since the cylinder is inscribed in the cone, its height will be equal to the height of the cone, which is 12 inches.
To apply the similar triangle property, we can draw the line OO'. The triangles AOO' and ABC are similar. This means that we can find the dimensions of the cylinder by determining the ratio of corresponding sides.
In triangle AOO', the ratio of corresponding sides is:
AO' / AO = CO / CB
Since CO = 12 inches and CB = 5 inches, the ratio becomes:
AO' / AO = 12 / 5
Now, we know that AO = radius of the cone = 5 inches. Let's substitute this into the ratio:
AO' / 5 = 12 / 5
Cross-multiplying, we get:
AO' = (12/5) * 5 = 12 inches
So, we have found that AO' (which is equal to the radius of the cylinder, r) is 12 inches.
Now, we need to find the height of the cylinder, which is h.
In triangle ABC, the ratio of corresponding sides is:
BC / AB = CO / AO
Since BC = 12 inches and AB = 5 inches, the ratio becomes:
12 / 5 = 12 / AO
Cross-multiplying, we get:
AO = (12/5) * 12 = 28.8 inches
So, we have found that AO (which is equal to the height of the cylinder, h) is 28.8 inches.
Therefore, the dimensions of the right circular cylinder of greatest volume inscribed in the cone are:
- Radius (r) = 12 inches
- Height (h) = 28.8 inches