What is the slope of the line passing through the points (1, −5) and (4, 1)?

<img src="https://tex.z-dn.net/?f=%20express%20%5C%3A%20%20%5C%3A%20%5Csin%282x%29%20%20-%20%20%5Csqrt%7B3%7D%20%20%5Ccos%282x%2

gizmo_the_mogwai [7]

Answer:

α = π/3

γ = 2

x = -3π/4, -5π/12, π/4, 7π/12

Step-by-step explanation:

Easiest way to do this is in reverse.

γ sin(2x − α)

Angle sum/difference formula:

γ (cos α sin(2x) − sin α cos(2x))

Distribute:

γ cos α sin(2x) − γ sin α cos(2x)

Matching the coefficients:

γ cos α = 1

γ sin α = √3

Solve the system of equations. Divide to eliminate γ:

tan α = √3

α is between 0 and π/2, so:

α = π/3

γ = 2

sin(2x) − √3 cos(2x) = 1

Using the identity from before:

2 sin(2x − π/3) = 1

Solving:

sin(2x − π/3) = 1/2

2x − π/3 = π/6 + 2kπ or 5π/6 + 2kπ

2x = π/2 + 2kπ or 7π/6 + 2kπ

x = π/4 + kπ or 7π/12 + kπ

x is between -π and π, so:

x = -3π/4, -5π/12, π/4, 7π/12

7 0

3 years ago

3(x-4) = 2(-2x + 1) this is confusing me

nata0808 [166]

Answer:

x = 2

Step-by-step explanation:

Distribute

3 (−4) = 2 (−2+1)

3 − 12 = 2 (-2x +1)

Distrubute again

3x - 12 = -4x + 2

Add twelve to both sides of the equation

Simpfly

3x = -4x + 14

Add 4x to both sides of the equation

simpfly

7x = 14

Divide both sides of the equation by the same term

Simpfly

x = 2

5 0

2 years ago

Find the slant height l of the cone given the surface area S.

Oduvanchick [21]

9514 1404 393

Answer:

15 cm

Step-by-step explanation:

The formula for the surface area of the cone is ...

A = πr² +πrL = πr(r +L)

The radius of this cone is half its diameter so is (12 cm)/2 = 6 cm. Then the slant height L can be found from ...

126π cm² = π(6 cm)(6 cm +L)

21 cm = 6 cm +L . . . . . . . . . . . . . . divide by 6π cm

15 cm = L . . . . . . . . . . . . . . . . subtract 6 cm

The slant height is 15 centimeters.

6 0

2 years ago

Khan Academy

zvonat [6]

Part 1

Answer: -9

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

Explanation:

Let's focus on evaluating first. We need to look at the table and locate f(x) = 13. So we're looking in the f(x) row where 13 shows up. That's in the second to last column. Right above that we have x = 5

This means f(5) = 13 and The inverse undoes the f(x) function. So the input x and output y values swap places. This is why we're reading the table in reverse.

We can replace all of $f^{-1}$ $(13)$ with 5 to go from $f^{-1} (f^{-1} (13))$ to $f^{-1} (5)$

Then we repeat the process of using the table. Locate 5 in the f(x) row. This is in the last column. The value above that is x = -9.

So f(-9) = 5 and $f^{-1} (5)=-9$

Overall, $f^{-1} (f^{-1} (13))=-9$

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

Part 2

Answer: -13

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

Explanation:

Same idea as before. Locate 8 in the f(x) row. The value above this is x = -13

This means f(-13) = 8 and $f^{-1}(8) =-13$

7 0

2 years ago

What is the total depreciation (at 10% rate) on a blower purchased 3 years ago for $499?

GrogVix [38]

$\bf \qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &499\\ r=rate\to 10\%\to \frac{10}{100}\dotfill &0.10\\ t=\textit{elapsed time}\dotfill &3\\ \end{cases} \\\\\\ A=499(1-0.10)^3\implies A=499(0.9)^3\implies A=363.771 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{\textit{total depreciation 499 - 363.771}}{135.229}~\hfill$

6 0

2 years ago