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3 years ago

Triangle abc is an equilateral triangle with ac =6; find the area of this triangle.

1 answer:

7 0

The area of an equilateral triangle of side length "s" is
.. A = (√3)/4*s^2

Your triangle will have an area of
.. A = (√3)/4*6^2 = 9√3 . . . . . square units

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Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 9t + 9 cot(t/2), [π/4, 7π/4]

agasfer [191]

Answer:

the absolute maximum value is 89.96 and

the absolute minimum value is 23.173

Step-by-step explanation:

Here we have cotangent given by the following relation;

$cot \theta =\frac{1 }{tan \theta}$ so that the expression becomes

f(t) = 9t +9/tan(t/2)

Therefore, to look for the point of local extremum, we differentiate, the expression as follows;

f'(t) = $\frac{\mathrm{d} \left (9t +9/tan(t/2) \right )}{\mathrm{d} t}$ = $\frac{9\cdot sin^{2}(t)-\left (9\cdot cos^{2}(t)-18\cdot cos(t)+9 \right )}{2\cdot cos^{2}(t)-4\cdot cos(t)+2}$

Equating to 0 and solving gives

$\frac{9\cdot sin^{2}(t)-\left (9\cdot cos^{2}(t)-18\cdot cos(t)+9 \right )}{2\cdot cos^{2}(t)-4\cdot cos(t)+2} = 0$

$t=\frac{4\pi n_1 +\pi }{2} ; t = \frac{4\pi n_2 -\pi }{2}$

Where n $_i$ is an integer hence when n₁ = 0 and n₂ = 1 we have t = π/4 and t = 3π/2 respectively

Or we have by chain rule

f'(t) = 9 -(9/2)csc²(t/2)

Equating to zero gives

9 -(9/2)csc²(t/2) = 0

csc²(t/2) = 2

csc(t/2) = ±√2

The solutions are, in quadrant 1, t/2 = π/4 such that t = π/2 or

in quadrant 2 we have t/2 = π - π/4 so that t = 3π/2

We then evaluate between the given closed interval to find the absolute maximum and absolute minimum as follows;

f(x) for x = π/4, π/2, 3π/2, 7π/2

f(π/4) = 9·π/4 +9/tan(π/8) = 28.7965

f(π/2) = 9·π/2 +9/tan(π/4) = 23.137

f(3π/2) = 9·3π/2 +9/tan(3·π/4) = 33.412

f(7π/2) = 9·7π/2 +9/tan(7π/4) = 89.96

Therefore the absolute maximum value = 89.96 and

the absolute minimum value = 23.173.

7 0

4 years ago

What the answer please help marking brainliest if it's correct and explain

kolezko [41]

Answer: The answer is C or Median

Step-by-step explanation:

8 0

3 years ago

PLZ HELP ME HURRY PWEASE!!!!!!!!!

Anna71 [15]

Answer:

1.5.80 miles

2.9.80ft

3.56 inches

4.5.20 units

5. 13 units

6.5.40 ft

Step-by-step explanation:

⇒These questions are on the Pythagorean relationship

⇒General expression is ;

a²+b²=c² where a and b are values for the legs and c is the hypotenuse

1. Isaiah waked 3 miles north and 5 miles east.

Finding the distance from starting point we apply the equation;

a²+b²=c² where a=5 miles and b=3miles

5²+3²=c²

25+9=c²

34=c²

c=√34 = 5.80 miles

2.The ladder is leaning against the wall to form the hypotenuse side of 10 ft.The distance from the base to the ladder forms side a= 2ft. The question requires you to calculate the height b.

a²+b²=c²

2²+b²=10²

4+b²=100

b²=100-4

b=√96

b=9.80 ft

3. The television has a diagonal of 70 inches to represent side c and a height of 42 inches to represent the side a. The question requires you to find the width of the television which represents the side b

a²+ b²=c²

42²+b²=70²

b²=70²-42²

b=√3136

c=56 inches

4. An equilateral triangle is one that has all its sides and angles equal. Here you have to introduce a perpendicular line from the apex point of the triangle to the base and get the distance a.The using the hypotenuse and the distance a, you can find the height/b.

c= 6 units and a=6/2= 3 units

a²+b²=c² 3²+ b²= 6²

9+b²=36

b²=36-9

b=√27 =5.20 units

5.In this question, you are given the height and the width of the right triangle thus apply the relationship as;

a²+b²=c²..........where a=5 and b=12

5²+12²=c²

25+144=c²⇒⇒⇒c=√169 =13 units

6. In this question, the height of the plant represents side b and the distance of the base from where the string is tied is side b.You are to find the length of the string which represents the hypotenuse side.

a²+b²=c²

5²+2²=c²

25+4=c²

29=c²

√29=c

c=5.40 ft

5 0

3 years ago

Analyze the diagram below and complete the instructions to follow.<br><br> Find a, b and c

Feliz [49]

Answer/Step-by-step explanation:

✔️Using trigonometric ratio, find b:

Reference angle = 45

opposite = 8

Hypotenuse = b

Thus,

Sin(45) = 8/b

b = 8/sin(45)

b = 8/(1/√2) (sin 45 = 1/√2)

b = 8 × √2/1

b = 8√2

✔️Find a using trigonometric ratio:

Reference angle = 60°

Hypotenuse = b = 8√2

Adjacent = a

Therefore,

Cos(60) = a/8√2

8√2*cos(60) = a

8√2*½ = a

4√2 = a

a = 4√2

✔️Find c using Pythagorean Theorem:

c² = b² - a²

c² = (8√2)² - (4√2)²

c² = 128 - 32

c² = 96

c = √96

c = √(16*6)

c = 4√6

5 0

3 years ago

Which two ratios form a proportion? A) 1 : 3 and 6 : 3 B) 1 : 3 and 3 : 6 C) 3 : 1 and 9 : 3 D) 3 : 1 and 3 : 9

tamaranim1 [39]

Answer:

Step-by-step explanation:

you have to set them up as a fraction and see if they are an equivalent fraction and if they are thats the proportion

4 0

3 years ago

Read 2 more answers