Mathematics: f(x)= 3sqrt x+11 Find the inverse of f(x)

01.08.2020

f(x)= 3sqrt x+11
Find the inverse of f(x)

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Mathematics

1. in abc, c is a right angle and bc= 11. if the measure of angle b= 30degrees, find ac. a) (11sqrt3)/3 b) 3sqrt/11 c)11/2 d)11sqrt3/2 2.which of the following angles has a reference angle of pi/8? a)-51pi/8 b)11pi/8 c)-33pi/8 d)27pi/8 3.your town’s public library is building a new wheelchair ramp to its entrance. by law, the maximum angle of incline for the ramp is 4.76 degrees. the ramp will have a vertical ride of 2.5 ft. what is the shortest horizontal distance that the ramp can span? a)11.9ft b)30.0ft c)4.3ft d)19.1ft 4. find the exact value

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

Mathematics

Neil places £2000 in a bank account that pays 1.5% simple interest per year. How much interest will he earn in 6 years?

Step-by-step answer

P Answered by PhD

SI=(P*R*T)/100

P=2000

R=1.5

T=6

SI=(2000*1.5*6)/100

=(2000*9)/100

=180

Neil will earn interest of 180

Mathematics

You can buy jars of the same jam in 2 sizes large jar:440g for £1.54 small jar:340g for £1.26 By considering the price per gram, work out which jar is the best value for money. Working must be shown

Step-by-step answer

P Answered by PhD

Answer: 440 grams for 1.54 is the better value
Explanation:
Take the price and divide by the number of grams
1.54 / 440 =0.0035 per gram
1.26 / 340 =0.003705882 per gram
0.0035 per gram < 0.003705882 per gram

Mathematics

Anthony received $150 for his birthday this year. This was $10 less than half the amount he received last year. How much money did Anthony receive for his birthday last year?

Step-by-step answer

P Answered by PhD

The answer is in the image