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KatRina [158]
2 years ago
9

Math Correct answers only please!!

Mathematics
1 answer:
aksik [14]2 years ago
6 0

Answer:

See below for answers and explanations

Step-by-step explanation:

<u>Problem 1</u>

Rewrite and add each vector:

p=\langle40cos70^\circ,40sin70^\circ\rangle\\q=\langle50cos270^\circ,50sin270^\circ\rangle\\r=\langle60cos235^\circ,60sin235^\circ\rangle\\p+q+r=\langle40cos70^\circ+50cos270^\circ+60cos235^\circ,40sin70^\circ+50sin270^\circ+60sin235^\circ\rangle\\p+q+r\approx\langle-20.73,-61.56\rangle

Find the magnitude of the resulting vector:

||p+q+r||=\sqrt{x^2+y^2}\\||p+q+r||=\sqrt{(-20.73)^2+(-61.56)^2}\\||p+q+r||\approx64.96

Therefore, the best answer is D) 64.959

<u>Problem 2</u>

Think of the vectors like this:

t=\langle7cos240^\circ,7sin240^\circ\rangle\\u=\langle10cos30^\circ,10sin30^\circ\rangle\\v=\langle15cos310^\circ,15sin310^\circ\rangle

By adding the vectors, we have:

t+u+v=\langle7cos240^\circ+10cos30^\circ+15cos310^\circ,7sin240^\circ+10sin30^\circ+15sin310^\circ\rangle\\t+u+v\approx\langle14.8,-12.55\rangle

Since the direction of a vector is \theta=tan^{-1}(\frac{y}{x}), we have:

\theta=tan^{-1}(\frac{y}{x})\\\theta=tan^{-1}(\frac{-12.55}{14.8})\\\theta\approx-40.3^\circ

Don't forget to take into account that the resulting vector is in Quadrant IV since the horizontal component is positive and the vertical component is negative, so we will add 360° to our angle to get the result:

\theta=360^\circ+(-40.3)^\circ\\\theta=319.7^\circ

Therefore, the best answer is D) 320°

<u>Problem 3</u>

Again, rewrite the vectors and add them:

u=\langle30cos70^\circ,30sin70^\circ\rangle\\v=\langle40cos220^\circ,40sin220^\circ\rangle\\u+v=\langle30cos70^\circ+40cos220^\circ,30sin70^\circ+40sin220^\circ\rangle\\u+v\approx\langle-20.38,2.48\rangle

Using the direction formula:

\theta=tan^{-1}(\frac{y}{x})\\\theta=tan^{-1}(\frac{2.48}{-20.38})\\\theta\approx-6.94^\circ

As the vector is located in Quadrant II, we need to add 180° to our angle:

\theta=180^\circ+(-6.94)^\circ\\\theta=173.06^\circ

Therefore, the best answer is D) 173°

<u>Problem 4</u>

Using scalar multiplication:

-3u=-3\langle35,-12\rangle=\langle-105,36\rangle

Find the magnitude of the resulting vector:

||-3u||=\sqrt{x^2+y^2}\\||-3u||=\sqrt{(-105)^2+(36)^2}\\||-3u||=111

Find the direction of the resulting vector:

\theta=tan^{-1}(\frac{y}{x})\\\theta=tan^{-1}(\frac{36}{-105})\\\theta\approx-18.92^\circ

As the vector is located in Quadrant II, we need to add 180° to our angle:

\theta=180^\circ+(-18.92)^\circ\\\theta=161.08^\circ

Therefore, the best answer is C) 111; 161°

<u>Problem 5</u>

Using scalar multiplication:

5v=5\langle25cos40^\circ,25sin40^\circ\rangle=\langle125cos40^\circ,125sin40^\circ\rangle

Since the direction of the angle doesn't change in scalar multiplication, it only affects the magnitude, so the magnitude of the resulting vector would be ||5v||=125, making the correct answer C) 125; 200°

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aksik [14]

Answer:

The number of minutes advertisement should use is found.

x ≅ 12 mins

Step-by-step explanation:

(MISSING PART OF THE QUESTION: AVERAGE WAITING TIME = 2.5 MINUTES)

<h3 /><h3>Step 1</h3>

For such problems, we can use probability density function, in which probability is found out by taking integral of a function across an interval.

Probability Density Function is given by:

f(t)=\left \{ {{0 ,\-t

Consider the second function:

f(t)=\frac{e^{-t/\mu}}{\mu}\\

Where Average waiting time = μ = 2.5

The function f(t) becomes

f(t)=0.4e^{-0.4t}

<h3>Step 2</h3>

The manager wants to give free hamburgers to only 1% of her costumers, which means that probability of a costumer getting a free hamburger is 0.01

The probability that a costumer has to wait for more than x minutes is:

\int\limits^\infty_x {f(t)} \, dt= \int\limits^\infty_x {}0.4e^{-0.4t}dt

which is equal to 0.01

<h3>Step 3</h3>

Solve the equation for x

\int\limits^{\infty}_x {0.4e^{-0.4t}} \, dt =0.01\\\\\frac{0.4e^{-0.4t}}{-0.4}=0.01\\\\-e^{-0.4t} |^\infty_x =0.01\\\\e^{-0.4x}=0.01

Take natural log on both sides

ln (e^{-0.4x})=ln(0.01)\\-0.4x=ln(0.01)\\-0.4x=-4.61\\x= 11.53

<h3>Results</h3>

The costumer has to wait x = 11.53 mins ≅ 12 mins to get a free hamburger

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