Cynthia plans to build a treehouse that is 1/3 the size of Andrew's tree house. Cynthia plans to make the area of her tree house at least 13 square feet. Write and solve an inequality to find the area of Andrew's tree house. Let x be the area of Andrew's tree house. Describe how you know which tree house is larger without solving the inequality.

area = sum of parallel sides . height/2

= (5.9 + 16.3) 4.6/2

= 22.2 . 2.3

= 51.06 m²

so the answer is A

Answer:

(identity has been verified)

Step-by-step explanation:

Verify the following identity:

tan(θ) (cot(θ)^2 + 1)/(tan(θ)^2 + 1) = cot(θ)

Multiply both sides by tan(θ)^2 + 1:

tan(θ) (cot(θ)^2 + 1) = ^?cot(θ) (tan(θ)^2 + 1)

(cot(θ)^2 + 1) tan(θ) = tan(θ) + cot(θ)^2 tan(θ):

tan(θ) + cot(θ)^2 tan(θ) = ^?cot(θ) (tan(θ)^2 + 1)

cot(θ) (tan(θ)^2 + 1) = cot(θ) + cot(θ) tan(θ)^2:

tan(θ) + cot(θ)^2 tan(θ) = ^?cot(θ) + cot(θ) tan(θ)^2

Write cotangent as cosine/sine and tangent as sine/cosine:

sin(θ)/cos(θ) + sin(θ)/cos(θ) (cos(θ)/sin(θ))^2 = ^?cos(θ)/sin(θ) + cos(θ)/sin(θ) (sin(θ)/cos(θ))^2

(sin(θ)/cos(θ)) + (cos(θ)/sin(θ))^2 (sin(θ)/cos(θ)) = cos(θ)/sin(θ) + sin(θ)/cos(θ):

cos(θ)/sin(θ) + sin(θ)/cos(θ) = ^?(cos(θ)/sin(θ)) + (cos(θ)/sin(θ)) (sin(θ)/cos(θ))^2

(cos(θ)/sin(θ)) + (cos(θ)/sin(θ)) (sin(θ)/cos(θ))^2 = cos(θ)/sin(θ) + sin(θ)/cos(θ):

cos(θ)/sin(θ) + sin(θ)/cos(θ) = ^?cos(θ)/sin(θ) + sin(θ)/cos(θ)

The left hand side and right hand side are identical:

Answer: (identity has been verified)

By using the Pythagorean trigonometric identities and substituting the expressions of tan θ, sec θ, and csc θ, we can simplify the given expression to prove that tan θ (1 + cot2 θ) / (1 + tan2 θ) equals cot θ.

To prove that tan θ ( 1 + cot2 θ ) / ( 1 + tan2 θ ) = cot θ, we can use trigonometric identities. Recall the Pythagorean identity which states that cot2 θ + 1 = csc2 θ and tan2 θ + 1 = sec2 θ. Using these identities, we can rewrite the expression on the left side of the equation:

tan θ ( 1 + cot2 θ ) / ( 1 + tan2 θ ) = tan θ * csc2 θ / sec2 θ

Since sec θ = 1/cos θ and csc θ = 1/sin θ, and remembering that tan θ = sin θ / cos θ, we substitute these into the expression:

tan θ * csc2 θ / sec2 θ = (sin θ / cos θ) * (1/sin2 θ) / (1/cos2 θ)

With simplification, the sin2 θ in the numerator and denominator cancel out, as do the cos2 θ terms, leaving us with:

cos θ / sin θ = cot θ

Thus, the original expression simplifies to cot θ.

Answer:

Step-by-step explanation:

0.79 x 8 =

6.72

15.21 - 6.72 =

Answer =

8.49

Answer:

364 inches cubed

Step-by-step explanation:

Answer:

364 inches cubed

Step-by-step explanation:

Ⓗⓘ ⓣⓗⓔⓡⓔ

Well, the formula for volume is L*H*W

L=4 in, H=13 in, W=7 in

4*13*7=364

(っ◔◡◔)っ ♥ Hope this helped! Have a great day! :) ♥

Answer:D

Step-by-step explanation:

Multiply the cost of the house by 0.18.

Answer:

We also know that If two angles are complementary then  their sum is equal to 90°. So, two angles ∠1 & ∠2 are complementary.

Step-by-step explanation:

As BA is perpendicular to BC. So, angle ∠ABC is 90°.

We also know that If two angles are complementary then  their sum is equal to 90°. As ∠ABC is the sum of the ∠1 & ∠2. In other words, ∠1 & ∠2 are complementary. So, two angles ∠1 & ∠2 are complementary.

Lets prove this.

Given: BA ⊥ BC

Prove: ∠1 & ∠2 are complementary.

Statement                                                       Reasons

1. ∠1 & ∠2 are complementary                      1. Given

2. ∠1 + ∠2 = 90°                                             2. Def. of complementary angles

Answer:

x= 1÷4

Step-by-step explanation:

2x - 6x + 1 =0

-4x + 1= 0

-4x = -1

Final answer:

Using the quadratic formula, the real solutions of the equation [tex]x^2 - 6x + 1 = 0[/tex]are x = 3 + 2√2 and x = 3 − 2√2 since the discriminant is positive, indicating two real solutions.

Explanation:

To find all real solutions of the quadratic equation  [tex]x^2 - 6x + 1 = 0[/tex], we can use the quadratic formula, which is x = −b ± √(b2 − 4ac) / (2a) where a, b, and c are coefficients from the equation in the form [tex]ax^2 + bx + c = 0[/tex]. For our equation, a = 1, b = −6, and c = 1. Let's apply these values into the formula:

Calculate the discriminant: (−6)2 − 4(1)(1) = 36 − 4 = 32.

Since the discriminant is positive, there are two real solutions.

Calculate the solutions: x = (6 ± √32) / (2 × 1) = (6 ± 4√2) / 2 = 3 ± 2√2.

The real solutions of the equation are x = 3 + 2√2 and x = 3 − 2√2.

Answer:

Part 1) John’s situation is modeled by a linear equation (see the explanation)

Part 2)  [tex]y=100x+300[/tex]

Part 3) [tex]\$12,300[/tex]

Part 4) [tex]\$2,700[/tex]

Part 5) Is a exponential growth function

Part 6) [tex]A=6,000(1.07)^{t}[/tex]

Part 7) [tex]\$11,802.91[/tex]

Part 8)  [tex]\$6,869.40[/tex]

Part 9) Is a exponential growth function

Part 10) [tex]A=5,000(e)^{0.10t}[/tex]    or  [tex]A=5,000(1.1052)^{t}[/tex]

Part 11)  [tex]\$13,591.41[/tex]

Part 12) [tex]\$6,107.01[/tex]

Part 13)  Natalie has the most money after 10 years

Part 14)  Sally has the most money after 2 years

Step-by-step explanation:

Part 1) What type of equation models John’s situation?

Let

y ----> the total money saved in a jar

x ---> the time in months

The linear equation in slope intercept form

y=mx+b

The slope is equal to

[tex]m=\$100\ per\ month[/tex]

The y-intercept or initial value is

[tex]b=\$300[/tex]

so

[tex]y=100x+300[/tex]

therefore

John’s situation is modeled by a linear equation

Part 2) Write the model equation for John’s situation

see part 1)

Part 3) How much money will John have after 10 years?

Remember that

1 year is equal to 12 months

so

[tex]10\ years=10(12)=120 months[/tex]

For x=120 months

substitute in the linear equation

[tex]y=100(120)+300=\$12,300[/tex]

Part 4) How much money will John have after 2 years?

Remember that

1 year is equal to 12 months

so

[tex]2\ years=2(12)=24\ months[/tex]

For x=24 months

substitute in the linear equation

[tex]y=100(24)+300=\$2,700[/tex]

Part 5) What type of exponential model is Sally’s situation?

we know that    

The compound interest formula is equal to  

[tex]A=P(1+\frac{r}{n})^{nt}[/tex] 

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest  in decimal

t is Number of Time Periods  

n is the number of times interest is compounded per year

in this problem we have  

[tex]P=\$6,000\\ r=7\%=0.07\\n=1[/tex]

substitute in the formula above

[tex]A=6,000(1+\frac{0.07}{1})^{1*t}\\ A=6,000(1.07)^{t}[/tex]

therefore

Is a exponential growth function

Part 6) Write the model equation for Sally’s situation

see the Part 5)

Part 7) How much money will Sally have after 10 years?

For t=10 years

substitute  the value of t in the exponential growth function

[tex]A=6,000(1.07)^{10}=\$11,802.91[/tex] 

Part 8) How much money will Sally have after 2 years?

For t=2 years

substitute  the value of t in the exponential growth function

[tex]A=6,000(1.07)^{2}=\$6,869.40[/tex]

Part 9) What type of exponential model is Natalie’s situation?

we know that

The formula to calculate continuously compounded interest is equal to

[tex]A=P(e)^{rt}[/tex] 

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest in decimal  

t is Number of Time Periods  

e is the mathematical constant number

we have  

[tex]P=\$5,000\\r=10\%=0.10[/tex]

substitute in the formula above

[tex]A=5,000(e)^{0.10t}[/tex]

Applying property of exponents

[tex]A=5,000(1.1052)^{t}[/tex]

 therefore

Is a exponential growth function

Part 10) Write the model equation for Natalie’s situation

[tex]A=5,000(e)^{0.10t}[/tex]    or  [tex]A=5,000(1.1052)^{t}[/tex]

see Part 9)

Part 11) How much money will Natalie have after 10 years?

For t=10 years

substitute

[tex]A=5,000(e)^{0.10*10}=\$13,591.41[/tex]

Part 12) How much money will Natalie have after 2 years?

For t=2 years

substitute

[tex]A=5,000(e)^{0.10*2}=\$6,107.01[/tex]

Part 13) Who will have the most money after 10 years?

Compare the final investment after 10 years of John, Sally, and Natalie

Natalie has the most money after 10 years

Part 14) Who will have the most money after 2 years?

Compare the final investment after 2 years of John, Sally, and Natalie

Sally has the most money after 2 years

Answer:

Proof in the explanation

Step-by-step explanation:

Trigonometric Equalities

Those are expressions involving trigonometric functions which must be proven, generally working on only one side of the equality

For this particular equality, we'll use the following equation

[tex]\displaystyle cos(x-y)=cos\ x\ cos\ y+sin\ x\ sin\ y[/tex]

The equality we want to prove is  

[tex]\displaystyle arccos\ \sqrt{\frac{2}{3}}-arccos\left(\frac{1+\sqrt{6}}{2\sqrt{3}}\right)=\frac{\pi}{6}[/tex]  

Let's set the following variables:

[tex]\displaystyle x=arccos\ \sqrt{\frac{2}{3}},\ y=arccos(\frac{1+\sqrt{6}}{2\sqrt{3}})[/tex]

And modify the first variable:

[tex]\displaystyle x=arccos\ \frac{\sqrt{6}}{3}}=>\ cos\ x= \frac{\sqrt{6}}{3}}[/tex]

Now with the second variable

[tex]\displaystyle y=arccos\ \frac{1+\sqrt{6}}{2\sqrt{3}}=>cos\ y=\frac{1+\sqrt{6}}{2\sqrt{3}}=\frac{\sqrt{3}+3\sqrt{2}}{6}[/tex]

Knowing that

[tex]sin^2x+cos^2x=1[/tex]

We compute the other two trigonometric functions of X and Y

[tex]\displaystyle sin \ x=\sqrt{1-cos^2\ x}=\sqrt{1-(\frac{\sqrt{6}}{3})^2}=\sqrt{1-\frac{6}{9}}=\frac{\sqrt{3}}{3}[/tex]

[tex]\displaystyle sin\ y=\sqrt{1-cos^2y}=\sqrt{1-\frac{(\sqrt{3}+3\sqrt{2})^2}{36}}}[/tex]

[tex]\displaystyle sin\ y=\sqrt{\frac{36-(3+6\sqrt{6}+18)}{36}}=\sqrt{\frac{15-6\sqrt{6}}{36}}[/tex]

Computing

[tex]15-6\sqrt{6}=(3-\sqrt{6})^2[/tex]

Then

[tex]\displaystyle sin\ y=\frac{3-\sqrt{6}}{6}[/tex]

Now we replace all in the first equality:

[tex]\displaystyle cos(x-y)=\frac{\sqrt{6}}{3}.\frac{\sqrt{3}+3\sqrt{2}}{6}+\frac{\sqrt{3}}{3}.\frac{3-\sqrt{6}}{6}[/tex]

[tex]\displaystyle cos(x-y)=\frac{3\sqrt{2}+6\sqrt{3}}{18}+\frac{3\sqrt{3}-3\sqrt{2}}{18}[/tex]

[tex]\displaystyle cos(x-y)=\frac{9\sqrt{3}}{18}=\frac{\sqrt{3}}{2}=cos\ \pi/6[/tex]

Thus, proven  

The circumference of a circle is the distance around the Outside of the circle.

The circumference of found by multiplying PI by the diameter of the circle.

Answer:

Option 2 (B) 2/3

Step-by-step explanation:

Got it correct

Answer:

45.7%

Step-by-step explanation:

16/14*40%=45.7%

The correct statement is that the profit is 60% if it had been sold for $16.

The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.Triangle?

Given

When an article was sold for $14 the profit was 40%.

When an article was sold for $14 the profit was 40%.

Then, the cost of the article will be $10.

If the article is sold for $16.

Then profit will be

[tex]\rm Profit = \dfrac{selling\ price \ - \ cost\ price}{cost\ price}*100\\\\\rm Profit = \dfrac{16-10}{10}*100\\\\\rm Profit = 60 \%[/tex]

Thus, the profit will be 60%.

More about the percentage link is given below.

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Answer:

y=-3/4x+3

Step-by-step explanation:

3x+4y=12

4y=12-3x

4y=-3x+12

y=-3/4x+12/4

y=-3/4x+3

Answer:

The answer is figure a :-)

The reason why is because all sides of figure A are equivalent.

Answer:

_____________________________________________

Step-by-step explanation:

The calculation of take home pay is the actual amount calculation of how much you will be credited on a job done well.

Step-by-step explanation:

Before calculation, things to be taken care of are -

Now knowing these much in exact form of amount, we may proceed to calculation -

Answer:

The quotient is 345.5

Step-by-step explanation:

Check image

Final answer:

To find the quotient of 1,382 divided by 4, divide each place value separately. The quotient is 343 with a remainder of 2.

Explanation:

To find the quotient of 1,382 divided by 4, we divide the thousands, hundreds, tens, and ones place separately. Starting from the leftmost digit, we divide 1,382 by 4.

4 divided by 1 is 0 with a remainder of 4. Bring down the next digit 3. So, we have 43. Next, divide 43 by 4. 4 divided by 43 is 10 with a remainder of 3. Bring down the next digit 8. So, we have 38. Finally, divide 38 by 4. 4 divided by 38 is 9 with a remainder of 2.

Therefore, the quotient of 1,382 divided by 4 is 343 remainder 2.

Answer:

[tex]f(x) = (840)\times (1.05)^{x}[/tex] where f(x) is the bonus every year and x is in number of years

Step-by-step explanation:

The function that represents the situation where $840 annual bonus increases by 5% each year is given by

[tex]f(x) = (840)\times (1.05)^{x}[/tex] where f(x) is the bonus every year and x is in number of years

Final answer:

The function representing an $840 annual bonus that increases by 5% each year is f(x) = 840(1 + 0.05)ˣ

Explanation:

To write a function that represents the situation where an $840 annual bonus increases by 5% each year, we can use an exponential growth model.

The general form of an exponential growth function is f(x) = a(1 + r)ˣ, where a is the initial amount, r is the growth rate, and x is the number of time periods.

In this case, the initial bonus a is $840, the growth rate r is 5% or 0.05, and x represents the number of years. Thus, the function can be written as:

f(x) = 840(1 + 0.05)ˣ

Answer:

n = 5

Step-by-step explanation:

Step  1  :

Step  2  :

Pulling out like terms :

2.1     Pull out like factors :

  -4n + 30  =   -2 • (2n - 15)

Equation at the end of step  2  :

 (0 -  -6 • (2n - 15)) -  -30  = 0

Step  3  :

Step  4  :

Pulling out like terms :

4.1     Pull out like factors :

  12n - 60  =   12 • (n - 5)

Equation at the end of step  4  :

 12 • (n - 5)  = 0

Step  5  :

Equations which are never true :

5.1      Solve :    12   =  0

This equation has no solution.

A a non-zero constant never equals zero.

Solving a Single Variable Equation :

5.2      Solve  :    n-5 = 0

Add  5  to both sides of the equation :

                     n = 5

One solution was found :

                  n = 5

Processing ends successfully

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To solve this, you need to isolate/get the variable "n" by itself in the equation:

-3(-4n + 30) = -30        Divide -3 on both sides

[tex]\frac{-3(-4n+30)}{-3} =\frac{-30}{-3}[/tex]     [two negative signs cancel each other out and become positive]

-4n + 30 = 10       Subtract 30 on both sides

-4n + 30 - 30 = 10 - 30

-4n = -20          Divide -4 on both sides to get "n" by itself

[tex]\frac{-4n}{-4}=\frac{-20}{-4}[/tex]

n = 5

PROOF

-3(-4n + 30) = -30       Substitute/plug in 5 into "n"

-3(-4(5) + 30) = -30

-3(-20 + 30) = -30      Simplify what's inside the parentheses  [PEMDAS}

-3(10) = -30

-30 = -30

Answer:

C(g) = 2.19g ; 2.5

Step-by-step explanation:

Question 1

Since the cost is $2.19 per gallon. For every gallon, the cost will increase by $2.19. Hence, the cost per gallon C(g) is 2.19g.

Question 2

f(1.5) = 3(1.5) - 2

= 4.5 - 2

= 2.5

Answer:

1. C(g) = 2.98g

2. 2.5

Step-by-step explanation:

1. 1 gallon cost $2.98

Therefore y gallon will cost = y2.98 ie C(g) = 2.98g

2. F(x) = 3x — 2

F(1.5) = 3x1.5 — 2 = 4.5 — 2 = 2.5

Answer:

The given equation is H = -16t^2 + 64t + 60, where t is the elapsed time in seconds. This equation represents the height, H, of an object thrown upward from the ground with an initial velocity of 64 ft/s.    

Step-by-step explanation:

1. The term -16t^2 represents the effect of gravity on the object. Since the coefficient is negative, it indicates that the object is moving upward against the force of gravity. The square of the time, t^2, shows that the effect of gravity increases as time passes.  2. The term 64t represents the initial velocity of the object. The coefficient 64 indicates that the object was thrown upward with an initial velocity of 64 ft/s. The time, t, shows the effect of the initial velocity on the height.  3. The constant term 60 represents any additional height above the ground at the start. It could be the height from which the object was thrown or any elevation from the ground.  By plugging different values of t into the equation, you can find the corresponding heights at different times. For example, if you substitute t = 0, the equation becomes H = -16(0)^2 + 64(0) + 60 = 60. This means that at the start (t = 0), the object is at a height of 60 feet above the ground.  To find the maximum height reached by the object, we need to determine the vertex of the parabolic equation. The vertex is given by the formula t = -b/(2a), where a and b are the coefficients of t^2 and t, respectively.  In this case, a = -16 and b = 64. Substituting these values into the formula, we get t = -64/(2(-16)) = 2 seconds. This means that the object reaches its maximum height after 2 seconds.  To find the maximum height, substitute t = 2 into the equation: H = -16(2)^2 + 64(2) + 60 = 64 feet. Therefore, the object reaches a maximum height of 64 feet above the ground after 2 seconds.  I hope this explanation helps you understand the meaning of the given equation and how to interpret it in the context of elapsed time and height. Let me know if this helped!

Answer:

b

Step-by-step explanation:

Answer: [tex]m=-21[/tex]

Step-by-step explanation:

In order to solve the exercise, you need to remember the following properties:

1. Addition property of equality:

If [tex]a=b[/tex], then [tex]a+c=b+c[/tex]

2. Subtraction property of equality:

If [tex]a=b[/tex], then [tex]a-c=b-c[/tex]

3. Divison property of equality:

If [tex]a=b[/tex], then [tex]\frac{a}{c}=\frac{b}{c}[/tex]

4. Multiplication property of equality:

If [tex]a=b[/tex], then [tex]a*c=b*c[/tex]

Then, given the following equation:

[tex]9+\frac{m}{3}=2[/tex]

You need to followw these steps in order solve for "m":

- Apply the Subtraction property of equality subtracting 9 from both sides of the equation:

[tex]9+\frac{m}{3}-9=2-9\\\\\frac{m}{3}=-7[/tex]

- Apply the Multiplication property of equality multiplying both sides of the equation by 3. Then, you get:

[tex](3)(\frac{m}{3})=(-7)(3)\\\\m=-21[/tex]

Answer:

  4 and 5

Step-by-step explanation:

  20 is greater than 4² = 16, and less than 5² = 25. So, the square root of 20 is between 4 and 5.

___

Your calculator will tell you that √20 ≈ 4.472, so is between 4 and 5.

Question:

In the right triangle shown, ∠B=60° and BC = 2√3

How long is AB?

Answer exactly, using a radical if needed.

The image of the triangle is attached below:

Answer:

The length of AB is [tex]4\sqrt{3}[/tex]

Explanation:

It is given that ∠B = 60° and BC = [tex]2\sqrt{3}[/tex]

To determine the length of AB, we shall use the cosine formula.

Because the value of the angle and its adjacent side is given and AB is the hypotenuse, we shall substitute the value of angle and adjacent side in the formula to find the value of AB.

Thus, the formula for [tex]\cos \theta[/tex] is given by

[tex]\cos \theta=\frac{a d j}{h y p}[/tex]

Where [tex]\theta=60[/tex] and [tex]adj= 2\sqrt{3}[/tex] and [tex]hyp=x[/tex]

Substituting these values in the formula, we get,

[tex]\cos 60=\frac{2 \sqrt{3}}{x}[/tex]

Interchanging, we get,

[tex]x=\frac{2 \sqrt{3}}{\cos 60}[/tex]

The value of [tex]cos 60 =\frac{1}{2}[/tex]

Substituting, we get,

[tex]x=\frac{2 \sqrt{3}}{\frac{1}{2} }[/tex]

[tex]x=4\sqrt{3}[/tex]

Thus, the value of x is [tex]4\sqrt{3}[/tex]

Hence, the length of the hypotenuse AB is [tex]4\sqrt{3}[/tex]

Answer:

1*56 8*7 ECT

Step-by-step explanation:

56 one time=56. 7 8 times = 56

In this problem, we use the Triangle Inequality Theorem to find that the minimum value for x, which would yield a valid triangle, is 27.67.

In this Mathematics problem, we're given a triangle with sides 9cm, 3x-20cm, 72cm, and 56cm. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. By doing this, we can set up an inequality to solve for x:

So, the minimum value for x that would yield a valid triangle is 27.67.

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Answer:

813 = 800 + 10 + 3 = (8 [tex]\times 10^{2}[/tex] ) + ( 1[tex]\times 10^1[/tex] ) + ( 3 [tex]\times 10^0[/tex] )  

Step-by-step explanation:

i) 813 = 800 + 10 + 3 = (8 [tex]\times 10^{2}[/tex] ) + ( 1[tex]\times 10^1[/tex] ) + ( 3 [tex]\times 10^0[/tex] )  

Answer:

x=-1/3, y=-13/3. (-1/3, -13/3).

Step-by-step explanation:

y=x-4

4x-y=3

------------

4x-(x-4)=3

4x-x+4=3

3x+4=3

3x=3-4

3x=-1

x=-1/3

y=-1/3-4

y=-1/3-12/3=-13/3

Answer: ([tex]-\frac{1}{3},-\frac{13}{3}[/tex]

Step-by-step explanation:

Substitut (x-4) into the equation of 4x-y=3

(Substitute what y equals into the equation

Make sure to keep parentheses!

That becomes 4x-(x-4)=3

Than you must distribute the negative to x and to -4

When you do it creates the equation of 4x-x+4=3

When combining like terms you get: 3x+4=3

Then solve                                                -4   -4

3x=-1

3    3               x=-1    

                           3

Than substitute the x in for the equation of y=x-4 to find what y equals! since you know that x equals -1/3 than subtract -1/3 - 4 to get

y= -13          

     3        

To complete the calculations, more data is needed. We only have the angle of elevation of the mother bird, we need one relevant data, as for example, the distance from Lisa to the tree. We'll assume it to be 20 feet.

Answer:

The mother bird is 47 feet above the ground

Step-by-step explanation:

Right Triangles

They are a special type of triangles that have an internal angle of 90°. In such conditions, the following trigonometric relationship is valid:

[tex]\displaystyle tan\theta=\frac{y}{x}[/tex]

Where [tex]\theta[/tex] is the angle of elevation, y is the height of the triangle and x is the horizontal distance to the vertical leg

We can easily find the height of the tree by solving for y

[tex]y=xtan\theta[/tex]

[tex]y=20tan67^o[/tex]

[tex]y=47\ feet[/tex]

The mother bird is 47 feet above the ground