How do I find the perimeter of similar triangles? (Geometry)

Intersection point of Y=logx and y=1/2log(x+1)

GalinKa [24]

Answer:

The intersection is $(\frac{1+\sqrt{5}}{2},\log(\frac{1+\sqrt{5}}{2})$ .

The Problem:

What is the intersection point of $y=\log(x)$ and $y=\frac{1}{2}\log(x+1)$ ?

Step-by-step explanation:

To find the intersection of $y=\log(x)$ and $y=\frac{1}{2}\log(x+1)$ , we will need to find when they have a common point; when their $x$ and $y$ are the same.

Let's start with setting the $y$ 's equal to find those $x$ 's for which the $y$ 's are the same.

$\log(x)=\frac{1}{2}\log(x+1)$

By power rule:

$\log(x)=\log((x+1)^\frac{1}{2})$

Since $\log(u)=\log(v)$ implies $u=v$ :

$x=(x+1)^\frac{1}{2}$

Squaring both sides to get rid of the fraction exponent:

$x^2=x+1$

This is a quadratic equation.

Subtract $(x+1)$ on both sides:

$x^2-(x+1)=0$

$x^2-x-1=0$

Comparing this to $ax^2+bx+c=0$ we see the following:

$a=1$

$b=-1$

$c=-1$

Let's plug them into the quadratic formula:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x=\frac{1 \pm \sqrt{(-1)^2-4(1)(-1)}}{2(1)}$

$x=\frac{1 \pm \sqrt{1+4}}{2}$

$x=\frac{1 \pm \sqrt{5}}{2}$

So we have the solutions to the quadratic equation are:

$x=\frac{1+\sqrt{5}}{2}$ or $x=\frac{1-\sqrt{5}}{2}$ .

The second solution definitely gives at least one of the logarithm equation problems.

Example: $\log(x)$ has problems when $x \le 0$ and so the second solution is a problem.

So the $x$ where the equations intersect is at $x=\frac{1+\sqrt{5}}{2}$ .

Let's find the $y$ -coordinate.

You may use either equation.

I choose $y=\log(x)$ .

$y=\log(\frac{1+\sqrt{5}}{2})$

The intersection is $(\frac{1+\sqrt{5}}{2},\log(\frac{1+\sqrt{5}}{2})$ .

6 0

3 years ago

Gracie, Mary, and Nancy each have a small collection of seashells. Gracie has 5 more than times the number of shells Mary has. N

Zielflug [23.3K]

Answer:

3/2 x +1 = 5/4x - 5

3/2 x+1 =5/4

x−5

Step 1: Subtract 5/4x from both sides.

3/2 x+1−5/4

x=5

4x−5−5/4x

1/4x+1=−5

Step 2: Subtract 1 from both sides.

1/4x+1−1=−5−1

1/4

x=−6

Step 3: Multiply both sides by 4.

4*(1/4x)=(4)*(−6)

x=−24 Mary has 24 shells

Answer:

-x=−24

x = 24 shells

Step-by-step explanation:

1. 2 3/4 x 16 = 44 + 6 = 50 shells x= 16

2 3/4 x 24 = 66 + 6 = 72 x = 24

We see the side workings in bold as if Mary has 24 and Nancy has 1x more

Then Nancy 24+24 = 48 where Gracie 5 x 24 = 120 and we have 72-48 = 24 and 120-72 = 48 being the same as Nancy. You can ignore or check the rest in explanation.

2 3/4 x 18 = 49 1/2 + 6 = 55 1/2 x= 18

2. 2 3/4 x 16 = 44 -4 = 40 shells x = 16 2 3/4 x 24 = 66 - 4 = 62 x = 24

2 3/4 x 18 = 49 1/2 - 4 = 45 1/2 x = 18

3. 2 3/4 x 16 = 44 + 4 = 48 shells x = 16 2 3/4 x 24 = 66 + 4 = 70 x = 24

2 3/4 x 18 = 49 1/2 + 4 =53 1/2 x = 18

4 0

3 years ago

Read 2 more answers

What us the length of the longer side?

julsineya [31]

The perimeter:
$P=2\cdot(3x+3)+2\cdot2x=6x+6+4x=10x+6\\\\P=146\ units\to10x+6=146\ \ \ |-6\\\\10x=140\ \ \ |:10\\\\x=14\ units$
The length of the longer side:
$3x+3=3\cdot14+3=42+3=45\ units$

3 0

3 years ago

WILL MARK BRAINLIEST!!!!<br><br> What is 40%, 1/4, 25%, 1/3 in least to greatest?

Ket [755]

The order is 1/4, 25%, 1/3, 40%

6 0

4 years ago

Find the mean, median, mode, and range of this data: 49, 49, 54, 55, 52, 49, 55. If necessary, round to the nearest tenth.

krok68 [10]

Answer:

Mean = 51.4.

Mode = 49.

Median = 52.

Range = 6.

Step-by-step explanation:

Mean = Sum of all observations / Number of observations.

Mean = (49+49+54+55+52+49+52)/7

Mean = 360/7

Mean = 51.4 (to the nearest tenth).

Mode = The most repeated values = 49 (repeated 3 times).

Range = Largest Value - Smallest Value = 55 - 49 = 6.

Median = The central value of the data.

First, arrange the data in the ascending order: 49, 49, 49, 52, 54, 55, 55.

It can be seen that the middle value is 52. Therefore, median = 52!!!

7 0

3 years ago