31.05.2023

Find the value of X round to the nearest degree x 9 19

. 4

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Mathematics
Step-by-step answer
P Answered by PhD
(A) Find the value of x. Round to the nearest tenth of a degree
Given that the distance of the ground is 9ft.
Length of the plank is 41 ft

In order to get the angle formed between the ground and the foot of the plank,

cos (theta) = ground / plank
cos (theta) = 9 / 41
cos (theta) = 0.21951...
theta = cos^-1(0.21951...)
theta = 77.3 degrees

(b) Find the height above the ground where the plank touches the wall. Round to the nearest tenth of a foot.

Height = square root (plank^2 - ground^2)
Height = square root (41^2 - 8^2)
Height = square root (1600)
Height = 40

(A) 77.1 Degrees
(B) 40 feet
Mathematics
Step-by-step answer
P Answered by PhD
Given:
bottom of the plank or ground is 9 feet from the wall
length of the plant is 41 ft
height of the wall is unknown.

Let us use the Pythagorean theorem.

a² + b² = c²
a² + (9ft)² = (41ft)²
a² + 81 ft² = 1,681 ft²
a² = 1,681 ft² - 81 ft²
a² = 1,600 ft²
√a² = √1,600 ft²
a = 40 ft

The height of the wall is 40 ft.
Mathematics
Step-by-step answer
P Answered by PhD
(A) Find the value of x. Round to the nearest tenth of a degree
Given that the distance of the ground is 9ft.
Length of the plank is 41 ft

In order to get the angle formed between the ground and the foot of the plank,

cos (theta) = ground / plank
cos (theta) = 9 / 41
cos (theta) = 0.21951...
theta = cos^-1(0.21951...)
theta = 77.3 degrees

(b) Find the height above the ground where the plank touches the wall. Round to the nearest tenth of a foot.

Height = square root (plank^2 - ground^2)
Height = square root (41^2 - 8^2)
Height = square root (1600)
Height = 40

(A) 77.1 Degrees
(B) 40 feet
Mathematics
Step-by-step answer
P Answered by PhD
(A) Find the value of x. Round to the nearest tenth of a degree
Given that the distance of the ground is 9ft.
Length of the plank is 41 ft

In order to get the angle formed between the ground and the foot of the plank,

cos (theta) = ground / plank
cos (theta) = 9 / 41
cos (theta) = 0.21951...
theta = cos^-1(0.21951...)
theta = 77.3 degrees

(b) Find the height above the ground where the plank touches the wall. Round to the nearest tenth of a foot.

Height = square root (plank^2 - ground^2)
Height = square root (41^2 - 8^2)
Height = square root (1600)
Height = 40

(A) 77.1 Degrees
(B) 40 feet
Mathematics
Step-by-step answer
P Answered by PhD
Given:
bottom of the plank or ground is 9 feet from the wall
length of the plant is 41 ft
height of the wall is unknown.

Let us use the Pythagorean theorem.

a² + b² = c²
a² + (9ft)² = (41ft)²
a² + 81 ft² = 1,681 ft²
a² = 1,681 ft² - 81 ft²
a² = 1,600 ft²
√a² = √1,600 ft²
a = 40 ft

The height of the wall is 40 ft.
Mathematics
Step-by-step answer
P Answered by Master

4.6cm

Step-by-step explanation:

A diagram showing the analysis from the question has been attached to this response. Kindly download it and view.

From the diagram,

=> We have three angles given

(i) the right angle, 90°

(ii) the angle opposite to side x, 50°

(iii) the angle opposite to side y, 40°

=> We also have one side given,

(i) the hypotenuse, of length 6cm

We can therefore apply the sine rule as follows;

\frac{sin 90^0}{6} = \frac{sin 50^0}{x} = \frac{sin 40^0}{y}           ---------------(i)

Now to get the value of x, we'll equation the first and second terms of equation (i) as follows;

\frac{sin 90^0}{6} = \frac{sin 50^0}{x}               [cross multiply]

xsin90^0 = 6 sin 50^0

x * 1 = 6 * 0.7660

x =4.596 cm = 4.6cm  [to the nearest tenth]

Therefore, the value of x is 4.6cm to the nearest tenth


What is the approximate value of x? Round to the nearest tenth. A right triangle is shown. Side x is
Mathematics
Step-by-step answer
P Answered by Master

4.6cm

Step-by-step explanation:

A diagram showing the analysis from the question has been attached to this response. Kindly download it and view.

From the diagram,

=> We have three angles given

(i) the right angle, 90°

(ii) the angle opposite to side x, 50°

(iii) the angle opposite to side y, 40°

=> We also have one side given,

(i) the hypotenuse, of length 6cm

We can therefore apply the sine rule as follows;

\frac{sin 90^0}{6} = \frac{sin 50^0}{x} = \frac{sin 40^0}{y}           ---------------(i)

Now to get the value of x, we'll equation the first and second terms of equation (i) as follows;

\frac{sin 90^0}{6} = \frac{sin 50^0}{x}               [cross multiply]

xsin90^0 = 6 sin 50^0

x * 1 = 6 * 0.7660

x =4.596 cm = 4.6cm  [to the nearest tenth]

Therefore, the value of x is 4.6cm to the nearest tenth


What is the approximate value of x? Round to the nearest tenth. A right triangle is shown. Side x is
Mathematics
Step-by-step answer
P Answered by PhD

QUESTION 16



If the demand and supply are equal, then we equate the two functions in p and solve for p.



That is



S(p)=D(p)


400 - 4p + + 0.00002 {p}^{4} = 2800 - 0.0012 {p}^{3}



We can rearrange to obtain,



0.00002 {x}^{4} + 0.0012 {x}^{3} - 4x - 2400 = 0



2 {p}^{2} + 1200 {p}^{3} - 400000p - 240000000 = 0



The real roots of this polynomial equation are:



p = - 118.26 \: p = 96.24



Since price can not be negative, we discard the negative value ,


p = 96.24


The correct answer for question 16 is C.



QUESTION 17



We were given the solution to this polynomial as



x=2,x=-4, x=1+3i



We need to recognize the presence of the complex root and treat it nicely.



There is one property about complex roots of polynomial equations called the complex conjugate property. According to this property, if


a + bi



is a solution to


p(x)


then the complex conjugate


a - bi


is also a root.



Since


x = 1 + 3i



is a solution then,



x = 1 - 3i


is also a solution.



Therefore we have



f(x) = (x - 2)(x + 4)(x - (1+3i))(x - (1 - 3i)



f(x) = ( {x}^{2} + 4x - 2x - 8)( {x}^{2} - (1 - 3i)x - (1 + 3i)x + (1 + 3i)(1 - 3i))



We expand to obtain,



( {x}^{2} + 4x - 2x - 8)( {x}^{2} - x + 3xi - x - 3xi + 1 + 9)


Note that:



{i}^{2} = - 1



f(x) = ( {x}^{2} + 2x - 8)( {x}^{2} - 2x + 10)



We now expand to obtain,



f(x) = {x}^{4} - 2 {x}^{3} + 10 {x}^{2} + 2 {x}^{3} - 4 {x }^{2} + 20x - 8 {x}^{2} + 16x - 80



We simplify further to obtain,



f(x) = {x}^{4} + 2 {x}^{2} + 36x - 80



The correct answer for question 17 is A.



QUESTION 18



If



-2i


is a zero of the polynomial,



f(x)=x^4-45x^2-196



then the complex conjugate



2i



is also a zero,



This means that ,


(x+2i), (x-2i)



are factors of the polynomial.



The product of these two factors,



(x+2i), (x-2i) =x^2-(2i)^2=x^2+4



is also a factor , so we use it to divide and get the remaining factors.



see diagram for long division.



The above polynomial can therefore factored completely as,



f(x)=(x^2+4)(x^2-49)



Applying our knowledge from difference of two squares, we obtain,



f(x)=(x+2i)(x-2i))(x-7)(x+7)



Hence all the zeroes of these polynomial can be found by setting



(x+2i)(x-2i))(x-7)(x+7)=0


This implies,



x=-2i,x=2i,x=7,x=-7



The correct answer for question 18 is B



QUESTION 19



We were asked to find the horizontal and vertical asymptote of



f(x)=\frac{2x^2+1}{x^2-1}



To find the horizontal asymptote, divide the term with the highest degree in the numerator by the term with the highest degree in the denominators. That is the horizontal asymptote is given by,



y=\frac{2x^2}{x^2}=2



For vertical asymptote, equate the denominator to zero and solve for x.



x^2-1=



\Rightarrow x=-1, x=1



None of the options is correct, so the correct answer for question 19 is A.



QUESTION 20



We are converting,



\frac{5\pi}{6}



to degrees .



To convert from radians to degrees, multiply by,



\frac{180\degree}{\pi}



That is,



\frac{5\pi}{6}=\frac{5\pi}{6} \times \frac{180\degree}{\pi}



We simplify to obtain,



\frac{5\pi}{6}=\frac{5}{1} \times \frac{180\degree}{1}=5\times30\degree =150\degree



The correct answer is B.



QUESTION 21



Recall the mnemonics, SOH CAH TOA



The sine ratio is given by,



\sin(B)=\frac{21}{75}=\frac{7}{25}



From the diagram,



\tan(B)=\frac{21}{72}=\frac{7}{24}



The correct answer is C.



QUESTION 22



From the above diagram, We can determine the value of x using the sine or cosine ratio, depending on where the 17 is placed.



Using the cosine ratio, we obtain,



\cos(58\degree)=\frac{17}{x}



We can simply switch positions to make x the subject.



x=\frac{17}{\cos(58\degree)}



x=\frac{17}{0.5299}



x=32.08



Hence the correct answer is A.



QUESTION 23



Coterminal angles have the same terminal sides.



To find coterminal angles, we keep adding or subtracting 360 degrees.



See diagram.



x=202\degree



is coterminal with



202\degree +360\degree =562\degree



or



202\degree -360\degree =-158\degree



The correct answer is D.



See the attached file for continuation.


16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
Mathematics
Step-by-step answer
P Answered by PhD

QUESTION 16



If the demand and supply are equal, then we equate the two functions in p and solve for p.



That is



S(p)=D(p)


400 - 4p + + 0.00002 {p}^{4} = 2800 - 0.0012 {p}^{3}



We can rearrange to obtain,



0.00002 {x}^{4} + 0.0012 {x}^{3} - 4x - 2400 = 0



2 {p}^{2} + 1200 {p}^{3} - 400000p - 240000000 = 0



The real roots of this polynomial equation are:



p = - 118.26 \: p = 96.24



Since price can not be negative, we discard the negative value ,


p = 96.24


The correct answer for question 16 is C.



QUESTION 17



We were given the solution to this polynomial as



x=2,x=-4, x=1+3i



We need to recognize the presence of the complex root and treat it nicely.



There is one property about complex roots of polynomial equations called the complex conjugate property. According to this property, if


a + bi



is a solution to


p(x)


then the complex conjugate


a - bi


is also a root.



Since


x = 1 + 3i



is a solution then,



x = 1 - 3i


is also a solution.



Therefore we have



f(x) = (x - 2)(x + 4)(x - (1+3i))(x - (1 - 3i)



f(x) = ( {x}^{2} + 4x - 2x - 8)( {x}^{2} - (1 - 3i)x - (1 + 3i)x + (1 + 3i)(1 - 3i))



We expand to obtain,



( {x}^{2} + 4x - 2x - 8)( {x}^{2} - x + 3xi - x - 3xi + 1 + 9)


Note that:



{i}^{2} = - 1



f(x) = ( {x}^{2} + 2x - 8)( {x}^{2} - 2x + 10)



We now expand to obtain,



f(x) = {x}^{4} - 2 {x}^{3} + 10 {x}^{2} + 2 {x}^{3} - 4 {x }^{2} + 20x - 8 {x}^{2} + 16x - 80



We simplify further to obtain,



f(x) = {x}^{4} + 2 {x}^{2} + 36x - 80



The correct answer for question 17 is A.



QUESTION 18



If



-2i


is a zero of the polynomial,



f(x)=x^4-45x^2-196



then the complex conjugate



2i



is also a zero,



This means that ,


(x+2i), (x-2i)



are factors of the polynomial.



The product of these two factors,



(x+2i), (x-2i) =x^2-(2i)^2=x^2+4



is also a factor , so we use it to divide and get the remaining factors.



see diagram for long division.



The above polynomial can therefore factored completely as,



f(x)=(x^2+4)(x^2-49)



Applying our knowledge from difference of two squares, we obtain,



f(x)=(x+2i)(x-2i))(x-7)(x+7)



Hence all the zeroes of these polynomial can be found by setting



(x+2i)(x-2i))(x-7)(x+7)=0


This implies,



x=-2i,x=2i,x=7,x=-7



The correct answer for question 18 is B



QUESTION 19



We were asked to find the horizontal and vertical asymptote of



f(x)=\frac{2x^2+1}{x^2-1}



To find the horizontal asymptote, divide the term with the highest degree in the numerator by the term with the highest degree in the denominators. That is the horizontal asymptote is given by,



y=\frac{2x^2}{x^2}=2



For vertical asymptote, equate the denominator to zero and solve for x.



x^2-1=



\Rightarrow x=-1, x=1



None of the options is correct, so the correct answer for question 19 is A.



QUESTION 20



We are converting,



\frac{5\pi}{6}



to degrees .



To convert from radians to degrees, multiply by,



\frac{180\degree}{\pi}



That is,



\frac{5\pi}{6}=\frac{5\pi}{6} \times \frac{180\degree}{\pi}



We simplify to obtain,



\frac{5\pi}{6}=\frac{5}{1} \times \frac{180\degree}{1}=5\times30\degree =150\degree



The correct answer is B.



QUESTION 21



Recall the mnemonics, SOH CAH TOA



The sine ratio is given by,



\sin(B)=\frac{21}{75}=\frac{7}{25}



From the diagram,



\tan(B)=\frac{21}{72}=\frac{7}{24}



The correct answer is C.



QUESTION 22



From the above diagram, We can determine the value of x using the sine or cosine ratio, depending on where the 17 is placed.



Using the cosine ratio, we obtain,



\cos(58\degree)=\frac{17}{x}



We can simply switch positions to make x the subject.



x=\frac{17}{\cos(58\degree)}



x=\frac{17}{0.5299}



x=32.08



Hence the correct answer is A.



QUESTION 23



Coterminal angles have the same terminal sides.



To find coterminal angles, we keep adding or subtracting 360 degrees.



See diagram.



x=202\degree



is coterminal with



202\degree +360\degree =562\degree



or



202\degree -360\degree =-158\degree



The correct answer is D.



See the attached file for continuation.


16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
16. photon lighting company determines that the supply and demand functions for its most popular lam
Mathematics
Step-by-step answer
P Answered by Specialist

1. The given triangle ABC, has a right angle at C, BC=11, and B=30\degree

\tan 30\degree=\frac{AC}{11}

AC=11\tan 30\degree

AC=\frac{11\sqrt{3}}{3}

Ans: A

2. The reference angle is the angle the terminal side makes with x-axis.

-\frac{33\pi}{8}=-4\frac{\pi}{8}

This implies that, -\frac{33\pi}{8} has a reference angle of \frac{\pi}{8}.

Ans: C

3. Let x be the shortest distance the ramp can span.

From the diagram; \tan (4.76\degree)=\frac{2.5}{x}

\implies x=\frac{2.5}{\tan (4.76\degree)}

\implies x=30.0ft

Ans:B

4. Use the Pythagorean identity: 1+\tan ^2 \theta=\sec^2 \theta.

If \cot \theta=-\frac{1}{2},then  \tan \theta=-2

\implies 1+2^2=\sec^2 \theta

\implies \sec^2 \theta=5

\implies \sec \theta=-\sqrt{5}, In QII, the secant ratio is negative.

Ans:C

5. We have \sin \frac{2\pi}{3}=\frac{\sqrt{3} }{2}

\cos \frac{\pi}{6}=\frac{\sqrt{3} }{2}

\cos \frac{\pi}{3}=\frac{1}{2}

\sin \frac{5\pi}{3}=-\frac{\sqrt{3} }{2}

\cos \frac{7\pi}{6}=-\frac{\sqrt{3} }{2}

\cos \frac{11\pi}{6}=\frac{\sqrt{3} }{2}

Ans:A and D

6.  The given function that is equivalent to f(x)=\sin x is f(x)=\cos (-x+\frac{\pi}{2}).

When we reflect the graph of  f(x)=\cos (x)  in the y-axis and shift it to the left by  \frac{\pi}{2} units, it coincides with graph of f(x)=\sin x.

Ans:C

7. The function y=\tan x is a one-to-one function on the interval [-\frac{\pi}{2},\frac{\pi}{2}]

When we restrict the domain of  y=\tan x on [-\frac{\pi}{2},\frac{\pi}{2}] it becomes an invertible function.

Ans: C

8. The given function is y=3\sin(4x-\pi)

The horizontal shift is given by \frac{C}{B}=\frac{\pi}{4}

The direction of the shift is to the right.

Ans:D

9.  \cos(-75\degree)=\cos(75\degree) by the symmetric property of even functions.

\cos(75\degree)=\cos(45\degree+30\degree)

\cos(75\degree)=\cos(45\degree) \cos30\degree-\sin(45\degree) \sin30\degree

\cos(75\degree)=\frac{\sqrt{2} }{2} \times \frac{\sqrt{3} }{2} -\frac{\sqrt{2} }{2} \times \frac{1}{2}

\cos(75\degree)=\frac{\sqrt{6}-\sqrt{2}}{4}

Ans: B

10. Recall the cosine rule: a^2=b^2+c^2-2bc\cos A

Let the angle measure opposite to the longest side be A, then a=19,b=17, and c=15.

\Rightarrow 19^2=17^2+15^2-2(17)(15)\cos A

\implies -153=-510\cos A

\implies \cos A=0.3

\implies A=\cos^{-1}(0.3)=73\degree

Ans:B

11.  We want to solve 2\sin(2x)\cos(x)-\sin(2x)=0 on the interval;

[-\frac{\pi}{2},\frac{\pi}{2}]

Factor:  \sin2(x)[2\cos(x)-1)=0

Either \sin(2x)=0 \implies x=0\frac{\pi}{2}

Or [2\cos x-1=0 This means that x=\frac{\pi}{3},-\frac{\pi}{3}

Therefore required solution is x=-\frac{\pi}{3},0,\frac{\pi}{3},\frac{\pi}{2}

Ans:D

12. Use the relation:r=\sqrt{x^2+y^2} and \theta=\tan^{-1}(\frac{y}{x})=

The given rectangular coordinate is (1,-2)

This implies that:r=\sqrt{1^2+(-2)^2}=\sqrt{5}

\theta=\tan^{-1}(\frac{-2}{1})= This means  \theta=116.6 or \theta=296.6

The polar forms are: -\sqrt{5},116.6 and \sqrt{5},296.6

Ans: B and C

13.  The polar equation that represents an ellipse is

r=\frac{2}{2-\sin \theta}.

When written in standard form; r=\frac{1}{1-0.5\sin \theta}.

The eccentricity is 0.5\:.

Therefore the r=\frac{2}{2-\sin \theta} is an ellipse.

Ans: B

14. The DeMoivre’s Theorem states that;

(\cos \theta+i\sin \theta)^n=\cos n\theta+i\sin n\theta

This implies that:

[2(\cos \frac{\pi}{9}+i\sin \frac{\pi}{9})]^3=2^3\cos 3\times \frac{\pi}{9}+i\sin 3\times \frac{\pi}{9})

[2(\cos \frac{\pi}{9}+i\sin \frac{\pi}{9})]^3=8(frac{2}{2})+i8(\frac{\sqrt{3}}{2})=4+4\sqrt{3}i

Ans: A

15. Let the initial point be (x,y), Then |v|=\sqrt{(-2-x)^2+(4-y)^2}.

If x=-8, and y=-4.

Then, |v|=\sqrt{(-2--8)^2+(4--4)^2}.

|v|=\sqrt{(-6)^2+(8)^2}=\sqrt{100}=10.

Ans: B

16. We find the dot product to see if it is zero.

u\bullet v=-6(7)+4(10)=-2

Since the dot product is not zero the vectors are not orthogonal

\theta=\cos ^{-1}(\frac{u\bullet v}{|u||v|})

\theta=\cos ^{-1}(-\frac{2}{2\sqrt{13}\times \sqrt{1149} }) =91.3\degree

Ans:B

17. Given v=5i+4j, w=2i-3j.

u=v+w

Add corresponding components

This implies u=(5i+4j)+(2i-3j)

u=(5i+2i+4j-3j)

u=7i+j

Ans:B

See attachment.


1. in abc, c is a right angle and bc= 11. if the measure of angle b= 30degrees, find ac. a) (11sqrt3
1. in abc, c is a right angle and bc= 11. if the measure of angle b= 30degrees, find ac. a) (11sqrt3

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