Mathematics, 25.09.2019 21:30 gymnastcaitlyn1

Solve using identities [0,2) cos2x=sinx

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We have:

$\cos 2x = 1 - 2 \sin^2 x$

Thus:

$1 - 2 \sin^2 x = \sin x$

Adding $2 \sin^2 x - 1$ gives:

$2 \sin^2 x + \sin x - 1 = 0$

Factoring gives:

$(2 \sin x + 1)(\sin x - 1) = 0$

Thus, we have:

$\sin x = \frac{-1}{2}$

or

$\sin x = 1$ .

Note that because $2 \pi 2$ , there will be no solutions

of the form $2n\pi + z$ for positive integer

and solution

. So we find the basic values of

giving $\sin x \in (\frac{-1}{2}, 1)$ .

For $\sin x = \frac{-1}{2}$ with positive

, $x \pi 2$ (this can be seen easily from the unit circle. But x < 2

, a contradiction. So we examine the case in which $\sin x = 1$ .

In this case, $x = \frac{\pi}{2}$ . So this is the only solution to this equation.

Substituting this back in, we have $\sin \frac{\pi}{2} = \cos \pi = 1$ , as desired. So this is a valid solution.

Thus, $x = \frac{\pi}{2}$ .

Answer from: Quest

where is the figure

Answer from: Quest

part a

answer: a reasonable domain is $0 \le n \le 10$ where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

=====================================

part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

=====================================

part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

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Solve using identities [0,2) cos2x=sinx...

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