What is the area of this triangle?
6 cm
11 cm

Part 1) the correct question is
If possible, find the area of the triangle defined by the following: b = 45, c = 20, A = 71°
The area of the triangle will be given by:
Area=1/2b*c sin A
Area=1/2*45*20*sin 71
Area=425.48 units²> 425.5 units²
=425.5 square units

The answer Part 1) is  425.5 square units


Part 2)Solve the triangle: b=12, c=10 B=45 . If it is not possible, say so.

applying the law of sines
b/sin B=c/sin C> b*sin C=c*sin B> sin C=(c/b)*sin B
sin C=(10/12)*sin 45> sin C=0.5892
C=arc sin(05892)> C=36.1°
A=180-(45+36.1)> A=98.9°
b/sin B=a/sin A> a=b*sin A/sin B> a=12*sin 98.9/sin 45
a=16.77> a=16.8

a=16.8 b=12  c=10
A=98.9° B=45° C=36.1°

the answer is the option
C) a=98.9, a=16.8, y=36.1

Part 3) A street light is mounted on a pole. The tip of the shadow of a man who is standing on a street a short distance from the pole has an angle of elevation to the top of his head of 28°. A woman standing in the opposite direction of the pole as the man was standing on the same street has a angle of elevation from the tip of her shadow to her head of 24°. If the two people are 20 feet apart, how far is the street light from the head of the woman?

Let 
A> angle of elevation of the shadow of a man> 28°
B> angle of elevation of the shadow of a woman> 24°
c=20 ft
C=180-(A+B)> C=180-(28+24)> 128°

applying the law of sines
20/sin C=a/sin A> a=20*sin 28/sin 128> a=11.92 ft> a=11.9 ft

the answer part 3) is the option
C)The street light is 11.9 feet from the head of the woman.

Part 4) Solve the triangle: a = 7, b = 9, C = 100°. If it is not possible, say so.

applying the law of cosines
c²=a²+b²-2*a*b*cos C> c²=7²+9²-2*7*9*cos 100°> c²=151.88
c=12.3

applying the law of sines
c/sin C=b/sin B> c*sin B=b*sin C> sin B=(b/c)*sin C
sin B=(9/12.3)*sin 100> sin B=0.7206
B=arc sin (0.7206)> B=46.1°
A=180-(100+46.1)> A=33.9°
so
a=7  b=9  c=12.3
A=33.9°  B=46.1°  C=100°

the answer part 4) is the option
D) A=34, B=46, C=12.3

Part 5)The triangle described is of the form SSA. Determine if there is no triangle possible, one triangle possible, or two triangles possible. If only one triangle is possible, solve the triangle. If either two triangles may be formed or no triangle may be formed, say so. The triangle is defined by: a=10, b=12, B=45
applying the law of sines
b/sin B=a/sin A> sin A=(a/b)*sin B---> (10/12)*sin 45--> 0.5892
A=arc sin (0.5892)> A=36.1°
C=180-(45+36.1)> C=98.9°
b/sin B=c/sin C> c=b*sin C/sin B> c=12*sin 98.9/sin 45
c=16.77> c=16.8
so
a=10  b=12  c=16.8
A=36.1°  B=45°  C=98.9°

the answer part 5) is the option
A)a=36.1, y=98.9, c=16.8

Part 6) Solve the triangle: a = 8, b = 10, c = 3. If it is not possible, say so.

applying the law of cosines
c²=a²+b²-2*a*b*cos C> cos C=[a²+b²-c²]/[2*a*b]
cos C=[8²+10²-3²]/[2*8*10]> 0.96875
C=arc cos (0.96875)> C=14.4°

applying the law of sines
c/sin C=b/sin B> sin B=(b/c)*sin C---> (10/3)*sin 14.4---> sin B=0.8290
B=arc sin(0.8290)> B=56°
A=180-(14.4+56)> A=109.6°
so
first triangle
A=109.6°  B=56°  C=14.4°

second triangle
C=14.4°
B=180-56=124°
A=180-(14.4+124)---> A=41.6

the answer part 6) is the option
B) A=41.4, B=124.2, C=14.4

Part 7)If possible, find the area of the triangle defined by the following: b = 11, c = 8, a = 28°.

The area of the triangle will be given by:
Area=1/2b*c sin A> Area=(1/2)*11*8*sin 28> 20.7 units²

the answer part 7) is the option
B) 20.7 square units

Part 8)For the function given, state the starting point for a sample period: ƒ(t) = 0.2sin (t − 0.3) + 0.1.

we know that
for t=0.3
ƒ(t) = 0.2sin (0.3 − 0.3) + 0.1> f(t)=0.2*sin (0)+0.1> 0.1
f(t)=0.1
so 
the point is
(0.3,0.1)

the answer Part 8) is the option
D) (0.3, 1)

Part 9)The triangle described is of the form SSA. Determine if there is no triangle possible, one triangle possible, or two triangles possible. If only one triangle is possible, solve the triangle. If either two triangles may be formed or no triangle may be formed, say so. The triangle is defined by: a=10, c=8, B=30

applying the law of cosines
b²=a²+c²-2*a*c*cos B> b²=10²+8²-2*10*8*cos 30> b²=25.44
b=5.04
applying the law of sines
b/sin B=a/sin A> sin A=(a/b)*sin B> (10/5.04)*sin 30
sin A=0.9920
A=arc sin (0.9920)> A=82.8°
C=180-(82.8+30)> C=67.2°

the first triangle dimensions are
a=10  b=5.04    c=8
A=82.8°  B=30°  C=67.2°

second triangle
A=180-82.8> 97.2°
B=30°
C=180-(97.2+30)> C=52.8

the answer part 9) is 
A) Two triangles may be formed

Part 1) the correct question is
If possible, find the area of the triangle defined by the following: b = 45, c = 20, A = 71°
The area of the triangle will be given by:
Area=1/2b*c sin A
Area=1/2*45*20*sin 71
Area=425.48 units²> 425.5 units²
=425.5 square units

The answer Part 1) is  425.5 square units


Part 2)Solve the triangle: b=12, c=10 B=45 . If it is not possible, say so.

applying the law of sines
b/sin B=c/sin C> b*sin C=c*sin B> sin C=(c/b)*sin B
sin C=(10/12)*sin 45> sin C=0.5892
C=arc sin(05892)> C=36.1°
A=180-(45+36.1)> A=98.9°
b/sin B=a/sin A> a=b*sin A/sin B> a=12*sin 98.9/sin 45
a=16.77> a=16.8

a=16.8 b=12  c=10
A=98.9° B=45° C=36.1°

the answer is the option
C) a=98.9, a=16.8, y=36.1

Part 3) A street light is mounted on a pole. The tip of the shadow of a man who is standing on a street a short distance from the pole has an angle of elevation to the top of his head of 28°. A woman standing in the opposite direction of the pole as the man was standing on the same street has a angle of elevation from the tip of her shadow to her head of 24°. If the two people are 20 feet apart, how far is the street light from the head of the woman?

Let 
A> angle of elevation of the shadow of a man> 28°
B> angle of elevation of the shadow of a woman> 24°
c=20 ft
C=180-(A+B)> C=180-(28+24)> 128°

applying the law of sines
20/sin C=a/sin A> a=20*sin 28/sin 128> a=11.92 ft> a=11.9 ft

the answer part 3) is the option
C)The street light is 11.9 feet from the head of the woman.

Part 4) Solve the triangle: a = 7, b = 9, C = 100°. If it is not possible, say so.

applying the law of cosines
c²=a²+b²-2*a*b*cos C> c²=7²+9²-2*7*9*cos 100°> c²=151.88
c=12.3

applying the law of sines
c/sin C=b/sin B> c*sin B=b*sin C> sin B=(b/c)*sin C
sin B=(9/12.3)*sin 100> sin B=0.7206
B=arc sin (0.7206)> B=46.1°
A=180-(100+46.1)> A=33.9°
so
a=7  b=9  c=12.3
A=33.9°  B=46.1°  C=100°

the answer part 4) is the option
D) A=34, B=46, C=12.3

Part 5)The triangle described is of the form SSA. Determine if there is no triangle possible, one triangle possible, or two triangles possible. If only one triangle is possible, solve the triangle. If either two triangles may be formed or no triangle may be formed, say so. The triangle is defined by: a=10, b=12, B=45
applying the law of sines
b/sin B=a/sin A> sin A=(a/b)*sin B---> (10/12)*sin 45--> 0.5892
A=arc sin (0.5892)> A=36.1°
C=180-(45+36.1)> C=98.9°
b/sin B=c/sin C> c=b*sin C/sin B> c=12*sin 98.9/sin 45
c=16.77> c=16.8
so
a=10  b=12  c=16.8
A=36.1°  B=45°  C=98.9°

the answer part 5) is the option
A)a=36.1, y=98.9, c=16.8

Part 6) Solve the triangle: a = 8, b = 10, c = 3. If it is not possible, say so.

applying the law of cosines
c²=a²+b²-2*a*b*cos C> cos C=[a²+b²-c²]/[2*a*b]
cos C=[8²+10²-3²]/[2*8*10]> 0.96875
C=arc cos (0.96875)> C=14.4°

applying the law of sines
c/sin C=b/sin B> sin B=(b/c)*sin C---> (10/3)*sin 14.4---> sin B=0.8290
B=arc sin(0.8290)> B=56°
A=180-(14.4+56)> A=109.6°
so
first triangle
A=109.6°  B=56°  C=14.4°

second triangle
C=14.4°
B=180-56=124°
A=180-(14.4+124)---> A=41.6

the answer part 6) is the option
B) A=41.4, B=124.2, C=14.4

Part 7)If possible, find the area of the triangle defined by the following: b = 11, c = 8, a = 28°.

The area of the triangle will be given by:
Area=1/2b*c sin A> Area=(1/2)*11*8*sin 28> 20.7 units²

the answer part 7) is the option
B) 20.7 square units

Part 8)For the function given, state the starting point for a sample period: ƒ(t) = 0.2sin (t − 0.3) + 0.1.

we know that
for t=0.3
ƒ(t) = 0.2sin (0.3 − 0.3) + 0.1> f(t)=0.2*sin (0)+0.1> 0.1
f(t)=0.1
so 
the point is
(0.3,0.1)

the answer Part 8) is the option
D) (0.3, 1)

Part 9)The triangle described is of the form SSA. Determine if there is no triangle possible, one triangle possible, or two triangles possible. If only one triangle is possible, solve the triangle. If either two triangles may be formed or no triangle may be formed, say so. The triangle is defined by: a=10, c=8, B=30

applying the law of cosines
b²=a²+c²-2*a*c*cos B> b²=10²+8²-2*10*8*cos 30> b²=25.44
b=5.04
applying the law of sines
b/sin B=a/sin A> sin A=(a/b)*sin B> (10/5.04)*sin 30
sin A=0.9920
A=arc sin (0.9920)> A=82.8°
C=180-(82.8+30)> C=67.2°

the first triangle dimensions are
a=10  b=5.04    c=8
A=82.8°  B=30°  C=67.2°

second triangle
A=180-82.8> 97.2°
B=30°
C=180-(97.2+30)> C=52.8

the answer part 9) is 
A) Two triangles may be formed

Reciprocals are:

1) 10 1/2 = 21/2

Reciprocal will be 2/21

2) 6 3/7 = 45/7

Reciprocal will be 7/45

3) 2 8/9 = 26/9

Reciprocal will be 9/26

4) 15 1/4 = 61/4

Reciprocal will be 4/61

5) 9 2/3 = 29/3

Reciprocal will be 3/29

6) 7 5/8 = 61/8

Reciprocal will be 8/61

Divide

7) 8/10 divide by 1 5/6

8/10 divide by 11/6

or 8/10 * 6/11

4/5* 6/11

24/55

8) 2 divides by 1 6/7

2 divides by 13/7

2 * 7/13

14/13

1 1/13

9) 3 3/5 divides by 2 1/4

18/5 divides by 9/4

18/5 * 4/9

8/5

1 3/5

10) 4 1/2 divides by 2 3/8

9/2 divides by 19/8

9/2 * 8/19

36/19

1 17/19

11) 5 5/6 divides by 3 1/6

35/6 divides by 19/6

35/6 * 6/19

35/19

1 16/19

12) 11/12 divides by 2 5/8

11/12 divides by 21/8

11/12 * 8/21

22/63

13) 1 9/13 divides by 3/8

22/13 divides by 3/8

22/13 * 8/3

176/39

4 20/39

14) 6 4/5 divides by 3 2/9

34/5 divides by 29/2

34/5 * 2/29

68/145

15) 9 2/3 divides by 6 8/9

29/3 divides by 62/9

87/62

1 25/69

Division problem

Width of patio = 5 2/3 feet

Area of patio = 36 5/6 sq. feet

Length will be: area /width

36 5/6 divides by 5 2/3

221/6 divides by 17/3

221/6 * 3/17

13/2

6 1/2 feet

Length is 6 1/2 feet, that means we can place the concrete patio on a 7 foot picnic table.

17) Area of mirror = 225 sq. inches

Width of mirror = 13 3/4 inches.

Length of mirror will be=area/ width

225 divides by 13 3/4

225 divides by 55/4

225 * 4/55

180/11

16 4/11

The width of mirror will be 16 4/11 feet. It is greater than 16 feet. Therefore mirror can't be fit in a space of 15 inches by 16 inches.

18) Total length of fabric = 16 1/5 yards

One fabric piece size= 5 2/5

Number of fabric pieces will be= total length/ 1 fabric length

16 1/5 divides by 5 2/5

81/5 divides by 27/5

81/5*5/27

3

There will be total 3 pieces of size 5 2/5 yards.

Reciprocals are:

1) 10 1/2 = 21/2

Reciprocal will be 2/21

2) 6 3/7 = 45/7

Reciprocal will be 7/45

3) 2 8/9 = 26/9

Reciprocal will be 9/26

4) 15 1/4 = 61/4

Reciprocal will be 4/61

5) 9 2/3 = 29/3

Reciprocal will be 3/29

6) 7 5/8 = 61/8

Reciprocal will be 8/61

Divide

7) 8/10 divide by 1 5/6

8/10 divide by 11/6

or 8/10 * 6/11

4/5* 6/11

24/55

8) 2 divides by 1 6/7

2 divides by 13/7

2 * 7/13

14/13

1 1/13

9) 3 3/5 divides by 2 1/4

18/5 divides by 9/4

18/5 * 4/9

8/5

1 3/5

10) 4 1/2 divides by 2 3/8

9/2 divides by 19/8

9/2 * 8/19

36/19

1 17/19

11) 5 5/6 divides by 3 1/6

35/6 divides by 19/6

35/6 * 6/19

35/19

1 16/19

12) 11/12 divides by 2 5/8

11/12 divides by 21/8

11/12 * 8/21

22/63

13) 1 9/13 divides by 3/8

22/13 divides by 3/8

22/13 * 8/3

176/39

4 20/39

14) 6 4/5 divides by 3 2/9

34/5 divides by 29/2

34/5 * 2/29

68/145

15) 9 2/3 divides by 6 8/9

29/3 divides by 62/9

87/62

1 25/69

Division problem

Width of patio = 5 2/3 feet

Area of patio = 36 5/6 sq. feet

Length will be: area /width

36 5/6 divides by 5 2/3

221/6 divides by 17/3

221/6 * 3/17

13/2

6 1/2 feet

Length is 6 1/2 feet, that means we can place the concrete patio on a 7 foot picnic table.

17) Area of mirror = 225 sq. inches

Width of mirror = 13 3/4 inches.

Length of mirror will be=area/ width

225 divides by 13 3/4

225 divides by 55/4

225 * 4/55

180/11

16 4/11

The width of mirror will be 16 4/11 feet. It is greater than 16 feet. Therefore mirror can't be fit in a space of 15 inches by 16 inches.

18) Total length of fabric = 16 1/5 yards

One fabric piece size= 5 2/5

Number of fabric pieces will be= total length/ 1 fabric length

16 1/5 divides by 5 2/5

81/5 divides by 27/5

81/5*5/27

3

There will be total 3 pieces of size 5 2/5 yards.

Area of the regular octagon is approximately 102ft²

Step-by-step explanation:

Given the radius and length of a side of regular octagon as shown;

radius = 6ft and length of side = 4.6ft

Area of a regular octagon is expressed as where a is the length of one side. Given a = 4.6ft

Area =

Area of the regular octagon is approximately 102ft²