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2 years ago

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Rationalization

rac{6}{ \sqrt{3} } " alt=" \frac{6}{ \sqrt{3} } " align="absmiddle" class="latex-formula">

1 answer:

rosijanka [135]2 years ago

7 0

Answer:

2√3

Step-by-step explanation:

To rationalize, we multiply the denominator and numerator by the surd at the base

We have it that;

6/√3 * 1

= 6/√3 * √3/√3

= 6 √3/3 = 2√(3

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Find dy/dx if f(x) = (x + 8)^3x.

vlabodo [156]

This is quite a complex problem. I wrote out a really nice solution but I can't work out how to put it on the website as the app is very poorly made. Still, I'll just have to type it all in...

Okay so you need to use a technique called logarithmic differentiation. It seems quite unnatural to start with but the result is very impressive.

Let y = (x+8)^(3x)

Take the natural log of both sides:

ln(y) = ln((x+8)^(3x))

By laws of logarithms, this can be rearranged:

ln(y) = 3xln(x+8)

Next, differentiate both sides. By implicit differentiation:

d/dx(ln(y)) = 1/y dy/dx

The right hand side is harder to differentiate. Using the substitution u = 3x and v = ln(x+8):

d/dx(3xln(x+8)) = d/dx(uv)

du/dx = 3

Finding dv/dx is harder, and involves the chain rule. Let a = x+ 8:

v = ln(a)
da/dx = 1
dv/da = 1/a

By chain rule:

dv/dx = dv/da * da/dx = 1/a = 1/(x+8)

Finally, use the product rule:

d/dx(uv) = u * dv/dx + v * du/dx = 3x/(x+8) + 3ln(x+8)

This overall produces the equation:

1/y * dy/dx = 3x/(x+8) + 3ln(x+8)

We want to solve for dy/dx, achievable by multiplying both sides by y:

dy/dx = y(3x/(x+8) + 3ln(x+8))

Since we know y = (x+8)^(3x):

dy/dx = ((x+8)^(3x))(3x/(x+8) + 3ln(x+8))

Neatening this up a bit, we factorise out 3/(x+8):

dy/dx = (3(x+8)^(3x-1))(x + (x+8)ln(x+8))

Well wasn't that a marathon? It's a nightmare typing that in, I hope you can follow all the steps.

I hope this helped you :)

8 0

3 years ago

Which of the following is defined by a line parallel to one side of a triangle

Katarina [22]

Answer:

Triangle Proportionality Theorem is defined by a line parallel to one side of a triangle divides the other two sides proportionally.

Step-by-step explanation:

Triangular Proportionality Theorem states that:

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Let us consider the triangle ΔABC as shown in attached figure.

If

$DE$ ║ $AC$

then

$\frac{BD}{DA} = \frac{BE}{EC}$

Therefore, Triangle Proportionality Theorem is defined by a line parallel to one side of a triangle divides the other two sides proportionally.

Keywords: Triangle Proportionality Theorem, triangle, line segment

Learn more about Triangle Proportionality Theorem from brainly.com/question/11827486

#learnwithBrainly

3 0

3 years ago

A rectangle has an area of 24 square centimeters. The width of the rectangle is 3 centimeters. What is the length of the rectang

brilliants [131]

Answer: length is 8

Step-by-step explanation:

its just 24 divided by 3 = 8

6 0

3 years ago

Read 2 more answers

matrenka [14]

Answer:

4. Slope of function B = -slope of function A

Step-by-step explanation:

Given:

Function A is given as:

$F(x)=-2x+1$

The above equation is of the form $y=mx+b$ , where $m$ represents slope of the line.

Therefore, on comparing the function A with the above standard form, er conclude that, slope of function A is -2.

Now, from the graph of function, we consider any two points on the graph and determine the slope of the line using the two points.

Let us consider the points $(x_1,y_1)=(1.5,0)\ and\ (x_2,y_2)=(3,3)$

Now, the slope of the line passing through these two points is given as:

$m_B=\frac{y_2-y_1}{x_2-x_1}=\frac{3-0}{3-1.5}=\frac{3}{1.5}=2$

Therefore, slope of function B is 2.

Therefore, the correct relation between the slopes of the two functions is that the slope of function B is negative of the slope of function A.

$m_B=-(m_A)=-(-2)=2$

5 0

3 years ago

PLEASE HELP!!!! TIMED EXAM

alexandr402 [8]

Answer:

The correct option is V = one-third (6) (4) (8).

Step-by-step explanation:

The volume of a rectangular pyramid is:

$V=\frac{1}{3}\ l\times b\times h$

Here,

V = volume

l = length

b = base

h = height

The information provided is:

l = 6 inches

b = 4 inches

h = 8 inches.

Compute the equation of the volume of a rectangular pyramid as follows:

$V=\frac{1}{3}\ l\times b\times h$

$=\frac{1}{3}\times 6\times 4\times 8$

Thus, the correct option is V = one-third (6) (4) (8).

4 0

3 years ago

Read 2 more answers