PLEASE HELP!!!!!!
Look at the triangle ABC.
A (4.5)
5
4
3
2
1
C (4.1)
B (2.1)
1 2 3
4 5
--5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
What is the length of the side AB of the triangle?
2
20
38

Answer:

r = 0.023104906

Step-by-step explanation:

Given half life = T = 30 yrs.

Decay constant = r.

Using the decay constant formula:

[tex]r=\frac{\ln2}{T}\\r=\frac{\ln2}{30}\\r=0.023104906[/tex]

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Answer:

[tex]\huge\boxed{-1}[/tex]

Step-by-step explanation:

=> 2 * 2 -5

Multiplying first according to DMAS rule

=> 4 - 5

=> -1

P.s. This is not an equation. This is an expression!

Answer:

[tex]\Huge \boxed{-1}[/tex]

Step-by-step explanation:

[tex]\Rightarrow 2 \times 2 -5[/tex]

Division and multiplication to be performed first.

[tex]\Rightarrow 4-5[/tex]

Addition and subtraction next.

[tex]\Rightarrow -1[/tex]

Answer:

90

Step-by-step explanation:

Answer:

90

Step-by-step explanation:

Here is the equation

[tex]20+3\times(7+4)+5+2\times(7+9)[/tex]

In the order of operations parentheses go first so we get

[tex]20+3\times11+5+2\times16[/tex]

Next we do the multiplication

[tex]20+33+5+32\\[/tex]

And finally we add them all up

[tex]20+33+5+32=90\\[/tex]

Thus, 90 is the answer of [tex]20+3\times(7+4)+5+2\times(7+9)[/tex] or [tex]20+3(7+4)+5+2(7+9)[/tex]

Answer:

7w

Step-by-step explanation:

Answer:B

Step-by-step explanation:

Answer:

Step-by-step explanation:

Hello, "the constant term has been written on the right side", it means that we add 18 to both sides to get.

[tex]x^2+3x-18=0\\\\x^2+3x=18\\\\\text{We can see the beginning of } (x+\dfrac{3}{2})^2 \\\\x^2+3x=(x+\dfrac{3}{2})^2-\dfrac{3^3}{2^2}=18\\\\(x+\dfrac{3}{2})^2=18+\dfrac{9}{4}=\dfrac{18*4+9}{4}=\dfrac{81}{4}[/tex]

Hope this helps.

Thank you.

Answer:

2.25.

Step-by-step explanation:

x^2 + 3x - 18 = 0

First, we need to write the constant on the right of the equation. So, we add 18 to both sides.

x^2 + 3x = 18.

Now, we find the number that will complete the square. It will be [tex](\frac{b}{2} )^2[/tex].

In this case, b = 3.

[tex](\frac{3}{2} )^2[/tex]

= (1.5)^2

= 2.25.

So, the number that will complete the square to solve the equation is 2.25, or 2 and 1/4, or 9/4.

Hope this helps!

Answer:

[-4+10+2. 0+25+3]

[10. 12]

[8. 28]

Option 3 is the solution

Answer:

131.2 gallons

Step-by-step explanation:

1)16/5= 3.2 gallons - of water are lost in 1 hour;

2)3.2*41=131.2 gallons

Answer:

I think it's 3

Step-by-step explanation:

because an=4 and the question is

an-1

=4-1

=3

Answer:

21/8 yds or 2 5/8 yds

Step-by-step explanation:

First turn 1 3/4 yards into an improper fraction so you can add it to 7/8 yards.

1 3/4 as an improper fraction is 7/4 yds

7/4 yds = 14/8 yds

14/8 yds + 7/8 yds = 21/8 yds

So the total amount of ribbon Amanda has is 21/8  yds or 2 5/8 yds

Answer and Step-by-step explanation:

The answer is 1722.1 x [tex]10^8[/tex]

#teamtrees #PAW (Plant And Water)

Answer:

1.72x10^10

Step-by-step explanation:

POSITIVE EXPONENT: means a number is huge

NEGATIVE EXPONENT: indicates a number is teeny-tiny

Answer:

13

Step-by-step explanation:

(3 - 2i)(3 + 2i)

Expand

(9 + 6i - 6i - 4i^2)

Add

(9 - 4i^2)

Convert i^2

i^2 = ([tex]\sqrt{-1}[/tex])^2 = -1

(9 - 4(-1))

Add

(9 + 4)

= 13

Answer:

13.

Step-by-step explanation:

(3 - 2i)(3 + 2i)

= (3 * 3) + (-2i * 3) + (2i * 3) + (-2i * 2i)

= 9 - 6i + 6i - 4[tex]\sqrt{-1} ^{2}[/tex]

= 9 - 4(-1)

= 9 + 4

= 13

Hope this helps!

Answer:

The angle of the slice is 63.64 degrees

Step-by-step explanation:

Here in this question, we are concerned with calculating the angle of the slice.

What we should know are as follows;

1. A pizza is a perfect circular shape

2. A 12-inch pizza means the diameter of the pizza is 12 inches and consequently its radius will be 12/2 = 6 inches

3. A slice of a pizza represents a sector of the circle( a sector is part of a circle bounded by 2 radii and an arc)

Mathematically, to calculate the angle of the slice, we simply use the formula for the area of a sector.

Area of sector = theta/360 * pi * r^2

where Area of sector = 20 square inches

theta is the center angle we are looking for

r is the radius of the pizza which is 6 inches

Plugging these values into the area of sector equation, we have

20 = theta/360 * 22/7 * 6^2

20 = theta/10 * 22/7

22 theta = 10 * 20 * 7

theta = 1400/22

theta = 63.64 degrees which is approximately 64 degrees to the nearest degree

Answer:

8 pounds of cheaper candy,

17.5 pounds of expensive candy

Step-by-step explanation:

Let's define some variables. Let's say the amount of pounds of candy that sells for $2.20/lb is x, and the $7.30 is y. Now we can write some equations!

x + y = 25.5

[tex]\frac{2.2x + 7.3y}{25.5} = 5.7[/tex]

We can start substitution. We can say that x = 25.5 - y. Plugging this into our second equation, we get:

y = 17.5

Plugging this in, we find that:

x = 8.

Answer:

C

Step-by-step explanation:

reciprocal of z=1/z

[tex]z=2(cos \frac{\pi }{4} +i sin\frac{\pi }{4} )=2e ^{i \frac{\pi } {4}\\\frac{1}{z}=\frac{1}{2e^{i \frac{\pi}{4} } }\\\frac{1}{z} =\frac{1}{2} e^{-i\frac{\pi}{4} } \\\frac{1}{z} (cos\frac{\pi}{4} -isin\frac{\pi}{4} ) \\\frac{1}{z}=\frac{1}{2} (\frac{\sqrt{2} }{2} -\frac{\sqrt{2} }{2} )\\\frac{1}{z} =\frac{\sqrt{2} }{4} -i \frac{\sqrt{2 } }{4}[/tex]

Answer:

the probability the die chosen was green is 0.9

Step-by-step explanation:

Given that:

A bag contains two six-sided dice: one red, one green.

The red die has faces numbered 1, 2, 3, 4, 5, and 6.

The green die has faces numbered 1, 2, 3, 4, 4, and 4.

From above, the probability of obtaining 4 in a single throw of a fair die is:

P (4  | red dice) = [tex]\dfrac{1}{6}[/tex]

P (4 | green dice) = [tex]\dfrac{3}{6}[/tex] =[tex]\dfrac{1}{2}[/tex]

A die is selected at random and rolled four times.

As the die is selected randomly; the probability of the first die must be equal to the probability of the second die = [tex]\dfrac{1}{2}[/tex]

The probability of two 1's and two 4's in the first dice can be calculated as:

= [tex]\begin {pmatrix} \left \begin{array}{c}4\\2\\ \end{array} \right \end {pmatrix} \times \begin {pmatrix} \dfrac{1}{6} \end {pmatrix} ^4[/tex]

= [tex]\dfrac{4!}{2!(4-2)!} ( \dfrac{1}{6})^4[/tex]

= [tex]\dfrac{4!}{2!(2)!} \times ( \dfrac{1}{6})^4[/tex]

= [tex]6 \times ( \dfrac{1}{6})^4[/tex]

= [tex](\dfrac{1}{6})^3[/tex]

= [tex]\dfrac{1}{216}[/tex]

The probability of two 1's and two 4's in the second  dice can be calculated as:

= [tex]\begin {pmatrix} \left \begin{array}{c}4\\2\\ \end{array} \right \end {pmatrix} \times \begin {pmatrix} \dfrac{1}{6} \end {pmatrix} ^2 \times \begin {pmatrix} \dfrac{3}{6} \end {pmatrix} ^2[/tex]

= [tex]\dfrac{4!}{2!(2)!} \times ( \dfrac{1}{6})^2 \times ( \dfrac{3}{6})^2[/tex]

= [tex]6 \times ( \dfrac{1}{6})^2 \times ( \dfrac{3}{6})^2[/tex]

= [tex]( \dfrac{1}{6}) \times ( \dfrac{3}{6})^2[/tex]

= [tex]\dfrac{9}{216}[/tex]

∴

The probability of two 1's and two 4's in both dies = P( two 1s and two 4s | first dice ) P( first dice ) + P( two 1s and two 4s | second dice ) P( second dice )

The probability of two 1's and two 4's in both die = [tex]\dfrac{1}{216} \times \dfrac{1}{2} + \dfrac{9}{216} \times \dfrac{1}{2}[/tex]

The probability of two 1's and two 4's in both die = [tex]\dfrac{1}{432} + \dfrac{1}{48}[/tex]

The probability of two 1's and two 4's in both die = [tex]\dfrac{5}{216}[/tex]

By applying  Bayes Theorem; the probability that the die was green can be calculated as:

P(second die (green) | two 1's and two 4's )  = The probability of two 1's and two 4's | second dice)P (second die) ÷ P(two 1's and two 4's in both die)

P(second die (green) | two 1's and two 4's )  = [tex]\dfrac{\dfrac{1}{2} \times \dfrac{9}{216}}{\dfrac{5}{216}}[/tex]

P(second die (green) | two 1's and two 4's )  = [tex]\dfrac{0.5 \times 0.04166666667}{0.02314814815}[/tex]

P(second die (green) | two 1's and two 4's )  = 0.9

Thus; the probability the die chosen was green is 0.9

Answer:

a) 6.00

b) 3.00

c) 1.50

Step-by-step explanation:

Sample error of the mean is expressed mathematically using the formula;

SE = σ /√n where;

σ  is the standard deviation and n is the sample size.

a) Given σ = 18, n = 9

Standard error of the mean = σ /√n

Standard error of the mean = 18/√9

Standard error of the mean = 18/3

Standard error of the mean = 6.00

b) Given σ = 18, n = 36

Standard error of the mean = σ /√n

Standard error of the mean = 18/√36

Standard error of the mean = 18/6

Standard error of the mean = 3.00

c) Given σ = 18, n = 144

Standard error of the mean = σ /√n

Standard error of the mean = 18/√144

Standard error of the mean = 18/12

Standard error of the mean = 3/2

Standard error of the mean = 1.50

Ali's average speed was 40 miles per hour.

What is an average speed?

The total distance traveled is to be divided by the total time consumed brings us the average speed.

How to calculate the average speed of Ali?

The total distance between the college from Ali's house is 135 miles.

She drove 2/3rd of the total distance in 135 minutes.

She drove =135*2/3miles

=90miles.

Ali can drive 90miles in 135 mins.

Therefore, her average speed is: 90*60/135 miles per hour.

=40 miles per hour.

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Answer:

[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV = \frac{\pi (17e^5 - 2)}{2}[/tex]

General Formulas and Concepts:
Calculus

Integration

Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method [Integration by Parts]:
[tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

Multivariable Calculus

Triple Integrals

Cylindrical Coordinate Conversions:

Spherical Coordinate Conversions:

Integral Conversion [Spherical Coordinates]:
[tex]\displaystyle \iiint_T {f( \rho, \phi, \theta )} \, dV = \iiint_T {\rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta[/tex]

Step-by-step explanation:

*Note:

Recall that φ is bounded by 0 ≤ φ ≤ 0.5π from the z-axis to the x-axis.

I will not show/explain any intermediate calculus steps as there isn't enough space.

Step 1: Define

Identify given.

[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV[/tex]

[tex]\displaystyle \text{Region E:} \ x^2 + y^2 + z^2 = 25 \ \text{bounded by first octant}[/tex]

Step 2: Integrate Pt. 1

Find ρ bounds.

Find θ bounds.

Find φ bounds.

Step 3: Integrate Pt. 2

We evaluate this spherical integral by using the integration rules, properties, and methods listed above:

[tex]\displaystyle \begin{aligned} \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 \int\limits^5_0 {e^{\rho} \rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta \\ & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 {\bigg[ (\rho^2 - 2 \rho + 2) e^{\rho} \sin \phi \bigg] \bigg| \limits^{\rho = 5}_{\rho = 0}} \, d\phi \, d\theta\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 {(17e^5 - 2) \sin \phi} \, d\phi \, d\theta \\& = \int\limits^{\frac{\pi}{2}}_0 {\bigg[ -(17e^5 - 2) \cos \phi \bigg] \bigg| \limits^{\phi = \frac{\pi}{2}}_{\phi = 0}} \, d\theta \\& = \int\limits^{\frac{\pi}{2}}_0 {17e^5 - 2} \, d\theta \\& = (17e^5 - 2) \theta \bigg| \limits^{\theta = \frac{\pi}{2}}_{\theta = 0} \\& = \frac{\pi (17e^5 - 2)}{2}\end{aligned}[/tex]

∴ the given integral equals [tex]\displaystyle \bold{\frac{\pi (17e^5 - 2)}{2}}[/tex].

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Topic: Multivariable Calculus

Unit: Triple Integrals Applications

Answer:  11 down and 9 right

Step-by-step explanation:

Slope IS rise over run where the top number of the fraction (numerator) determines the vertical distance --> positive is up, negative is down

and the bottom number of the fraction (denominator) determines the horizontal distance --> positive is right, negative is left.

Given slope = -11/9

the numerator is -11 so the "rise" is  DOWN 11 units

the denominator is 9 so the "run" is RIGHT 9 units

Answer:

The AP is 1, 11/2, 10, 29/2, 19, ....

Step-by-step explanation:

Let the first term be a and d be the common difference of the arithmetic progression.

ATQ, a+2d+a+6d=38, 2a+8d=38 and a+8d=37. Solving this, we will get a=1 and d=9/2. The AP is 1, 11/2, 10, 29/2, 19, ....

Answer:

[tex]\boxed{x=15}[/tex]

Step-by-step explanation:

[tex]\frac{3}{4} =\frac{x}{20}[/tex]

[tex]\sf Cross \ multiply.[/tex]

[tex]4 \cdot x = 20 \cdot 3[/tex]

[tex]4x=60[/tex]

[tex]\sf Divide \ both \ sides \ by \ 4.[/tex]

[tex]\frac{4x}{4} =\frac{60}{4}[/tex]

[tex]x=15[/tex]

Answer:

120 words

Step-by-step explanation:

as a ratio it's

2:180

In real life the longer you do something the more words you type

so since you can't get to 3 from 2 easily you go to 1 instead.

2:180

1:360

3:120

Answer:

a

See in the explanation

a-2.

Discrete

b-1.

Mean = 4.201

Standard Deviation = 2.069

b-2.

4.201

c.

Mean = 16.153

Standard Deviation = 8.079

Step-by-step explanation:

Given Data:

Number of Hours         Frequency               Amount Charged  

          1                                 16                                $3        

          2                                34                                 6  

          3                                51                                  12  

          4                                39                                 16  

          5                                34                                 21

          6                                 16                                 24

          7                                  9                                  27

          8                                 30                                 29

                                       ∑f = 229

a. Convert the information on the number of hours parked to a probability distribution:

The probability is calculated by dividing each frequency by 229. For example probability of Hour 1 is calculated as:

16 / 229 = 0.06987

This way all the hours probabilities are computed. The probability distribution is given below

Hours          Probability

   1                0.06987

   2               0.14847

   3               0.2227

   4               0.1703

   5               0.1485

   6               0.0699

   7               0.0393

   8               0.1310

   ∑                    1

a-2. Is this a discrete or a continuous probability distribution?

This is a discrete probability distribution as the probability of each hour of between 0 and​ 1 and the sum of all the probabilities of hours is 1.

b-1. Find the mean and the standard deviation of the number of hours parked.

First multiply each value of Number of hours by the corresponding frequency. Let x represents the number of hours and f represents frequency. Then:

Number of Hours Parked

fx

16

68

153

156

170

96

63

240

Now add the above computed products.

∑fx = 16+68+153+156+170+96+63+240  = 962

Compute Mean:

Now the formula to calculate mean:

Mean = Sum of the value / Number of value

         = ∑fx / ∑f

         = 962 / 229

Mean = 4.201

Compute Standard Deviation:

Let x be the Number of hours.        

Let f be the frequency

First calculate (x-x_bar) where x is each number of hours and x_bar is mean. The value of x_bar i.e. [tex]\frac{}{x}[/tex] = 4.201

For example for the Hour = 1 , and mean = 4.201

Then (x-[tex]\frac{}{x}[/tex]) = 1 - 4.201 = -3.201

So calculating this for every number of hour we get:

(x-[tex]\frac{}{x}[/tex])

-3.201

-2.201

-1.201

-0.201

0.799

1.799

2.799

3.799

Next calculate (x-[tex]\frac{}{x}[/tex])². Just take the squares of the above column (x-[tex]\frac{}{x}[/tex])

For example the first entry of below calculation is computed by:

 (x-[tex]\frac{}{x}[/tex])² = (-3.201 )² = 10.246401

  (x-[tex]\frac{}{x}[/tex])²

10.246401

4.844401

1.442401

0.040401

0.638401

3.236401

7.834401

14.432401

Next multiply each entry of  (x-[tex]\frac{}{x}[/tex])² with frequency f. For example the first entry below is computed by:

(x-[tex]\frac{}{x}[/tex])² * f = 10.246401  * 16 = 163.942416

(x-[tex]\frac{}{x}[/tex])² * f

163.942416

164.709634

73.562451

1.575639

21.705634

51.782416

70.509609

432.97203

Now the formula to calculate standard deviation is:

S = √∑(x-[tex]\frac{}{x}[/tex])² * f/n

Here

n = ∑f = 229

∑(x-[tex]\frac{}{x}[/tex])² * f  is the sum of all entries of (x-[tex]\frac{}{x}[/tex])² * f

∑(x-[tex]\frac{}{x}[/tex])² * f = 980.759829

S = √∑(x-[tex]\frac{}{x}[/tex])² * f/n

  = √980.759829  /  229

  = √4.2827940131004

  = 2.0694912449924

S = 2.069

b-2) How long is a typical customer parked?

That is the value of mean calculated in part b-1. Hence

Typical Customer Parked for 4.201 hours

c) Find the mean and the standard deviation of the amount charged.

First multiply each value of Amount Charged by the corresponding frequency. Let x represents the number of hours and f represents frequency. Then:

fx

48

204

612

624

714

384

243

870

Now add the above computed products.

∑fx = 48+204+612+624+714+384+243+870  = 3699

Compute Mean:

Now the formula to calculate mean:

Mean = Sum of the value / Number of value

         = ∑fx / ∑f

         = 3699 / 229

Mean = 16.153

Compute Standard Deviation:

Let x be the Amount Charged.        

Let f be the frequency.

First calculate (x-x_bar) where x is each value of Amount charged and x_bar is mean. The value of x_bar i.e. [tex]\frac{}{x}[/tex] = 16.153

For example for the Amount Charged = 3 , and mean = 16.153

Then (x-[tex]\frac{}{x}[/tex]) = 3 - 16.153 = -13.153

So calculating this for every number of hour we get:

(x-[tex]\frac{}{x}[/tex])

-13.153

-10.153

-4.153

-0.153

4.847

7.847

10.847

12.847

Next calculate (x-[tex]\frac{}{x}[/tex])². Just take the squares of the above column (x-[tex]\frac{}{x}[/tex])

For example the first entry of below calculation is computed by:

 (x-[tex]\frac{}{x}[/tex])² = (-13.153  )² = 173.001409

  (x-[tex]\frac{}{x}[/tex])²

173.001409

103.083409

17.247409

0.023409

23.493409

61.575409

117.657409

165.045409

Next multiply each entry of  (x-[tex]\frac{}{x}[/tex])² with frequency f. For example the first entry below is computed by:

(x-[tex]\frac{}{x}[/tex])² * f = 173.001409  * 16 =  

   (x-[tex]\frac{}{x}[/tex])² * f

2768.022544

3504.835906

879.617859

0.912951

798.775906

985.206544

1058.916681

4951.36227

∑(x-[tex]\frac{}{x}[/tex])² = 14947.65066

Now the formula to calculate standard deviation is:

S = √∑(x-[tex]\frac{}{x}[/tex])² * f/n

Here

n = ∑f = 229

∑(x-[tex]\frac{}{x}[/tex])² * f  is the sum of all entries of (x-[tex]\frac{}{x}[/tex])² * f

∑(x-[tex]\frac{}{x}[/tex])² * f = 14947.65066

S = √∑(x-[tex]\frac{}{x}[/tex])² * f/ ∑f

  = √65.273583668122

  = 8.0792068712295

S = 8.079

9514 1404 393

Answer:

  after 32 minutes

Step-by-step explanation:

They will both cross the start line together after each has completed a number of full laps in the same time that the other has completed an integer number of full laps. That time will be the least common multiple (LCM) of their lap times, 128 seconds and 120 seconds.

The difference of these times (8 seconds) is also a divisor of both times, so that value is the greatest common divisor (GCD) of the two lap times. The LCM can be figured as ...

  LCM(120, 128) = 120×128/GCD(120, 128)

  LCM = 120×128/8 = 1920 . . . . . seconds

Converting this to minutes, we get ...

  (1920 s)/(60 s/min) = 32 min

They will next cross the start line together after 32 minutes.

The domain exists on all real numbers i.e {x∈R}∈

The range exists all on real numbers except at y = 0

Domains are all input values of a function for which the function exists while ranges are all the output values for which the function exists.

Since the x-intercept of a function exists at where y = 0, this means that the point where a function does not have any x-intercepts are all other points on the graph except at y = 0.

The following statements are therefore true;

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Answer:

432

Step-by-step explanation:

x=1hr (6x)=6hours x= 72 hours through train so (6x)=72x6=432

Answer:

Step-by-step explanation:

f(x) = 3x + 5

g(x) = 4x² - 2

h(x) = x² - 3x + 1

f(x)+g(x)-h(x) means subtract h(x) from the sum of f(x) and h(x)

f(x)+g(x)-h(x) = 3x + 5 +( 4x² - 2) - (x² - 3x + 1)

Remove the brackets

That's

f(x)+g(x)-h(x) = 3x + 5 + 4x² - 2 - x² + 3x - 1

Group like terms

We have

f(x)+g(x)-h(x) = 4x² - x² + 3x + 3x + 5 - 2 - 1

Simplify

We have the final answer as

Hope this helps you

Hope this helps you

Answer:

20 gallons of the 40% solution, 60 gallons of the 80% solution

Step-by-step explanation:

Let x = the gallons of the 40% solution, and y = the gallons of a 80% solution. The first thing we want to do here is to convert each percentage into decimal form - including the ​70% solution mixture.

40% = 0.40,

80% = 0.80, respectively 70% = 0.70

As you can tell, 0.40 is associated with x gallons, 0.80 is associated with y gallons, and the mixture contains 0.70 [tex]*[/tex] 80 solution, as 0.70 is associated with 80. Therefore we can formulate the following expression,

0.40x + 0.80y = 0.70 [tex]*[/tex] 80

At the same time x + y = 80, as the solution ( mixture ) is present with 80 gallons. Isolating x, x = 80 - y. Let us plug that into our expression, solving for y, following by x gallons.

[tex]0.40\left(\:80\:-\:y\:\right)\:+\:0.80y\:=\:0.70\:\cdot \:80[/tex]

[tex]0.4\left(80-y\right)+0.8y=56[/tex] ( Multiply either side by 10 )

[tex]4\left(80-y\right)+8y=560[/tex] ( Expand )

[tex]320-4y+8y=560[/tex]

[tex]320+4y=560[/tex]

[tex]4y=240[/tex]

[tex]y = \frac{240}{4} = 60[/tex] ( Substitute to solve for x )

[tex]x = 80 - y = 80 - 60 = 20[/tex]

As you can see there are 20 gallons of the 40% solution, and 60 gallons of the 80% solution.