do now: express the area, a, of an equilateral triangle as a function of its sides
DESCRIPTION
Do Now: Express the area, A, of an equilateral triangle as a function of its sides. What is the relationship between the radius of the base and the height of a cone?. Let’s observe some things that are true about this diagram:. Congruent Triangles:. height. Similar Triangles:. - PowerPoint PPT PresentationTRANSCRIPT
1
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
Do Now: Express the area, A, of an equilateral triangle as a function of its sides.
1
2A b h
1 13
2 2A s s
213
4A s
13
2
Let b s
h s
13
2s
s
s
s
1
2s
1
2s
2
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
A B
C
D
E
water level
radius
Ex) A water tank has the shape of a right circular cone with its vertex pointing down. The radius, r, of the top is 2 ft and the height, h, of the tank is 8 ft. Express the volume of the water in the tank as a function of its depth.
FG
Let’s observe some things that are true about this diagram:
AABC DCGEFC FC
Congruent Triangles:
EABC FCGADC FC
Similar Triangles:
What is the relationship between the radius of the base and the height of a cone?
heig
ht r
h
radius r
height h
As a result, we may set up the following proportion:
3
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
water level
Ex) A water tank has the shape of a right circular cone with its vertex pointing down. The radius, r, of the top is 2 ft and the height, h, of the tank is 8 ft. Express the volume of the water in the tank as a function of its depth (height).
r
h
radius r
height h21
3V r h
PRIMARY SECONDARY
2
8
r
h
2
8
4
hr h
Write the radius in terms of h and substitute that radius into the volume equation to get the volume equation in terms of h as well.
2
1
3 4
hV h h
2
2
1
3 4
hV h h
31
3 16
hV h
31
48V h h
4
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
4. A water tank has the shape of a right circular cone with its vertex pointing downward. The radius of the top is 3 ft., and the height of the tank is 12 ft. Express the volume of the water in the tank as a function of its depth.
21
3V r hFunctions Worksheet #4-6
5
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
r
h
21
3V r h
PRIMARY SECONDARY
r h
21
3V h h h
Vertical Cross Sections: A vertical slicing of a cone creating 2 isos. right triangles.
45
45
60
5. The vertical cross sections of a conical water tank are isosceles right triangles. Express the volume of the water in the tank as a function of its depth.
The vertical cross sections of a cone are the triangles that are formed when you slice a cone vertical starting from the tip of the vertex; when you open the cone and look at it head-on you see, in this case, 2 Isosceles right triangles.
45
6
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
r
h
21
3V r h
PRIMARY SECONDARY
3h r
2
1
3 3
hV h h
Vertical Cross Sections: A vertical slicing of a cone creating 2 equilateral triangles.
30
60
60
3
hr h
31
9V h h
6. The vertical cross sections of a conical water tank are equilateral triangles. Express the volume of the water in the tank as a function of its depth.
60
2r2r
7
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
FINISH WORKSHEET: #7, 8, 9
8
AIM: VERTICAL CROSS SECTIONS OF CONES. HW: Functions Worksheet #4-6
REIVEW OF FUNCTION APPS:
A rectangle is bounded by the x-axis and the semi-circle
A Write the area of the rectangle as a function of x and determine the domain of the function.
236y x
66
6
x
y
RECTA L W 2
L y
W x
2RECTA xy
22 36RECTA x x
:0 6D x
What if the question said that the rectangle is inscribed in the entire circle?
2
2
L y
W x
x