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Answer:
The total surface area of the solid \(T\), formed by joining a cone and hemisphere with a 7 cm diameter each, can be calculated by summing the surface areas of both shapes.
To find the total surface area
(
total
)
(A
total
) of the solid
T , formed by joining a solid cone and a solid hemisphere, we need to calculate the surface areas of the cone and hemisphere separately and then add them together.
Given:
- Diameter of the base of the cone = 7 cm
- Diameter of the hemisphere = 7 cm
First, let's draw a diagram to represent the solid \(T\):
```
__________
/ \
/ \
/ \
/___________________\
```
Now, let's calculate the surface area of each component:
1. Surface area of the cone (\(A_{\text{cone}}\)):
- The formula for the surface area of a cone is
cone
=
cone
(
cone
+
cone
)
A
cone
=πr
cone
(r
cone
+l
cone
) , where
cone
r
cone
is the radius of the base and
cone
l
cone
is the slant height.
- Since the diameter of the cone's base is 7 cm, the radius
(
cone
)
7
/
2
=
3.5
(r
cone
)is7/2=3.5 cm.
- We need to find the slant height
(
cone
)
(l
cone
) . Using the Pythagorean theorem,
cone
=
cone
2
+
ℎ
cone
2
l
cone
=
r
cone
2
+h
cone
2
, where \(h_{\text{cone}}\) is the height of the cone.
- As the cone's height is not given, let's assume it's the same as the radius, \(h_{\text{cone}} = r_{\text{cone}} = 3.5\) cm.
- Substituting the values into the formula, we get
cone
=
×
3.5
×
(
3.5
+
3.
5
2
+
3.
5
2
)
.
A
cone
=π×3.5×(3.5+
3.5
2
+3.5
2
).
2. Surface area of the hemisphere
(
hemisphere
)
(A
hemisphere
) :
- The formula for the surface area of a hemisphere is
hemisphere
=
2
hemisphere
2
.
−
ℎ
ℎ
ℎ
ℎ
7
,
ℎ
(
hemisphere
)
7
/
2
=
3.5
.
A
hemisphere
=2πr
hemisphere
2
.−Sincethediameterofthehemisphereis7cm,theradius(r
hemisphere
)is7/2=3.5cm.
- Substituting the value into the formula, we get
hemisphere
=
2
×
3.
5
2
.
A
hemisphere
=2π×3.5
2
.
Finally, the total surface area
(
total
)
(A
total
) of the solid
T is the sum of the surface areas of the cone and hemisphere:
total
=
cone
+
hemisphere
A
total
=A
cone
+A
hemisphere
Substitute the calculated values to find the total surface area.
The question probably maybe: What is the total surface area of the solid \( T \), formed by joining a solid cone with a diameter of 7 cm at its base to a solid hemisphere with a diameter of 7 cm?