Diego wants to paint his room purple. He bought one gallon of purple paint that is 30% red paint and 70% blue paint. Diego wants to add more blue to the mix so that the paint mixture is 20% red, 80% blue.
1. How much blue paint should Diego add? Test the following possibilities: 0.2 gallons, 0.3 gallons, 0.4 gallons, 0.5 gallons.

0.2 gallons
0.2 gallons,

0.3 gallons
0.3 gallons,

0.4 gallons
0.4 gallons,

0.5 gallons
0.5 gallons,
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2. Write an equation in which x
represents the amount of paint Diego should add.


3. Check that the amount of paint Diego should add is a solution to your equation.

[tex]y = mx + b \\ y = nx + c \\ for \: the \: two \: lines \: to \: be \: perpendicular \\ m.n = - 1[/tex]

[tex]m = 3 \\ \\ 3n = - 1 \\ n = \frac{ - 1}{3} [/tex]

[tex]y = \frac{ - x}{3} + 2[/tex]

Answer:

see explanation

Step-by-step explanation:

3- digit numbers starting with 5 are

528 and 582

3- digit numbers starting with 2 are

258 and 285

3- digit numbers starting with 8 are

825 and 852

there are 6 possible 3- digit numbers

258 , 285 , 528 , 582 , 825 , 852

There are 7 ones on the left and only 2 on the right. The two sides need to be even, or balanced.

7 = 2 + r

r = 5

Hope this helps!

Using a calculator, the exponential regression equation for the data is:

y = 16615.21e^(-0.14x)

Using it, the estimate for the value of car after 11 years is $3,561.99.

To find the equation, we need to insert the points (x,y) in the calculator.

For this problem, the points are given as follows:

(0, 16700), (1, 14370), (2, 12520), (3, 10301), (4, 9871), (5, 7982), (6, 6984).

Inserting these points in the calculator, the equation is:

y = 16615.21e^(-0.14x)

Hence the value of the car after 11 years is given by:

y = 16615.21e^(-0.14 x 11) = $3,561.99.

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The greatest common factor for the following sets of numbers are as follows,

24, 34, 567 : 1

23, 36, 52, 11 : 1

37, 53, 72, 11², 13 : 1

1, 2, 3, 4, 6, 8, 12, and 24 make up the number 24.

1, 2, 17, and 34 make to the number 34.

1, 3, 7, 9, 21, 27, 63, 81, 189, and 567 make up the number 567.

Because of this, the case's greatest common factor is 1.

The elements of 11 are: 1, 11

The 23 factors are: 1, 23,

1, 2, 3, 4, 6, 9, 12, 18, and 36 make up the number 36.

The sum of 52's factors is: 1, 2, 4, 13, 26, and 52.

Therefore, 1 is the number that these numbers have the most in common, and thus, 1 is the greatest common factor here.

The elements of 13 are: 1, 13, and

These are the 37 factors: 1, 37

These are the components of 53: 1, 53

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 make up the number 72.

1, 11, and 121 make up the number 121.

Thus, the greatest common factor for 37, 53, 72, 121, 13 is again 1.

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Answer:

D

Step-by-step explanation:

Your equation would be 2x^2-3=y

This means that it should touch the y axis at -3, but since there is an exponent, it should be a parabola, not a straight line. Hence the answer is D

ADG is an equilateral triangle, then, line AD is equivalent to line AG.

From the information given, we have to prove that;

AD=AG

It is important to note that the triangle ADG is an equilateral triangle.

The properties of an equilateral triangle are;

Since ADG is an equilateral triangle, then, line AD is equivalent to line AG

Thus, since ADG is an equilateral triangle, then, line AD is equivalent to line AG.

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Answer:

question 4:

the smaller building is 262.31 metres tall

Step-by-step explanation:

see the diagram attached:

1. draw a diagram:

2. as you can see in the diagram, a right angled triangle has been created.

3. convert the known angle from degrees and minutes to degrees.

4.  Identify which trigonometry ratio to use:

sin = [tex]\frac{opposite}{hypotenuse}[/tex]

cos = [tex]\frac{adjacent}{hypotenuse}[/tex]

tan = [tex]\frac{opposite}{adjacent}[/tex]

5. insert known values into the equation:

6. Interpreting the question with new knowledge:

the height of building 2 is 262.31 metres.

The half-life of the substance is 1.94 years.

The exponential decay formula aids in determining the exponential drop, which is a rapid reduction over time. To calculate population decay, half-life, radioactivity decay, and other phenomena, one uses the exponential decay formula. F(x) = a [tex](1-r)^{x}[/tex] is the general form.

Here

a = the initial amount of substance

1-r is the decay rate

x = time span

The correct form of the equation is given as:

[tex]a=a_{0}[/tex]×[tex](0.7)^{t}[/tex]

where t is an exponent of 0.7 since this is an exponential decay of 1st order reaction

Now to solve for the half life, this is the time t in which the amount left is half of the original amount, therefore that is when:

a = 0.5 a0

Substituting this into the equation:

0.5 [tex]a_{0}=a_{0}[/tex]×[tex](0.7)^{t}[/tex]

0.5 = [tex](0.7)^{t}[/tex]

Taking the log of both sides:

t log 0.7 = log 0.5

t = log 0.5 / log 0.7

t = 1.94 years

The half life of the substance is 1.94 years.

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The solution to the equation 1/x =x+3/2x^2 is x = 0 and 3. Option C

In solving for the values of 'x' we need to:

Given the expression;

1/x =x+3/2x^2

Cross multiply

x ( x+ 3) = 2x² ( 1)

Expand the bracket

x² + 3x = 2x²

collect like terms and equate all the variables to zero, we have ;

2x² - x² - 3x = 0

We then subtract the like terms, we getv

x²- 3x = 0

Now, let's find the common factor

Factor out 'x':

x ( x - 3 ) = 0

Equate each of the factors to zero and find the value of 'x' for each

So,

x = 0

And

x - 3 = 0

x = 3

Thus, the solution to the equation 1/x =x+3/2x^2 is x = 0 and 3. Option C

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The extra amount that Mauro has to pay in federal taxes because he won the lottery would grossly amount to $336100.50. Option B

The term tax refers to money that the citizens of a country are mandatorily required to pay to the government from their income. This money is used for the provision of basic amenities for the people of the country.

We are able to understand from all the information provided in the question that that the extra amount that Mauro has to pay in federal taxes because he won the lottery would grossly amount to $336100.50.

Tax paid before lottery is;:

= $4386.25 + [($34500 - $31850) × 0.25]

= $4386.25 + ($2650 × 0.25)

= $4386.25 + $662.50

= $5048.75

Tax paid after lottery is:

= [($1000000 + $34500 - $349700) × 0.35)] + $101469.25

= $239680 + $101469.25

= $341149.25

Tax difference is:

= $341149.25 - $5048.75

= $336100.50

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Missing parts;

Mauro just won $1,000,000 from the lottery! despite his excitement in winning a tremendous amount of money, his brother (who is an accountant) warns him not to spend it all at once and hands him this tax rate schedule for single taxpayers. taxable income federal tax $0-$7,825 10% of amount $7825.01-$31,850 $725.50 plus 15% of amount over 7,825 $31,850.01-$77,100 $4,386.25 plus 25% of amount over 31,850 $77,100.01-$160,850 $15,698.75 plus 28% of amount over 77,100 $160,850.01-$349,700 $39,148.75 plus 33% of the amount over 160,850 $349,700.01 and up $101,469.25 plus 35% of amount over 349,700 using this chart, a taxpayer who makes $58,000 annually (line 3) will have to pay $4,386.25 plus 25% of the differene of 58,000 and 31,850 or 4,386.25 + (0.25)(58,000-31,850) = $10,923.75. before winning the lottery, mauro's annual income was $34,500. use the chart to determine mauro's federal tax before winning the lottery and after winning the lottery. how much extra does he have to pay in federal taxes because he won the lottery?

a. $5,048.75

b. $336,100.50

c. $341,149.25

d. $676,260.76

Answer:

16b > 94

Step-by-step explanation:

16b > 94

b > 5.875

Crystal needs 6 boxes to fit all 94 discs.

Answer:   C

Step-by-step explanation: We see that e is 6 for $2.70 and d is 12 for $6.00.  Half of 6 is 3 and 2.70 is less than 3 so the cheapest price can't be d. Then, we have 9 for 3.60. 3.60 divided by 9 is 0.40 per which is cheaper than a (0.42 per) so a and d can't be the cheapest. 3 for 1.38 means 0.46 per which is more expensive than a so it also cannot be b. Now we only have c and e left. We divide 2.70 by 6 and find out that the answer is 0.45 which is also more expensive than a. So, the answer is c.

Answer:

y-intecept: (0,14), x-intercept: (-8,0)

Step-by-step explanation:

Let us first solve for the y-intercept, solve the equation for y to set it to slope-intercept form.

[tex]-7x+4y=56\\4y=56+7x\\y=14+\frac{7}{4}x\\y=\frac{7}{4}x+14\\[/tex]

Now we have our y-intercept, 14

To solve for the x-intercept, all we have to do is set y to 0 and solve for x:

[tex]0=\frac{7}{4}x+14\\-14=\frac{7}{4}x\\-56=7x\\-8=x[/tex]

Therefore our x-intercept is -8.

To find the vector (x and y value) of the x intercept, we set the y value equal to 0, such as the vector of the x-intercept is equal to (-8,0).

For the y-intercept we follow a similar approach setting the x-value equal to 0, making it (0,14)

Answer:

C

Step-by-step explanation:

4 is a perfect square.  2 x 2.  You can pull the 2 out and everything else would be left under the square root symbol.

Answer:

2.625

Step-by-step explanation:

2 2/7 in decimal form is 2.2857

6 / 2.2857 = 2.625

The principal amount that must be deposited as a lump sum amount is = $16,666.7

The simple interest amount = $20,000

The interest rate = 12%

The time that is given= 10 years

Therefore the principle amount =?

Using the simple interest formula;

SI= P×T×R/100

Make P the subject of formula,

P = SI ×100/T×R

P= 20,000×100/10×12

P= 2,000,000/120

P= $16,666.7

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The volume of the given figure is 3617.3 km³

From the question, we are to calculate the volume of the given figure

The given figure is a cone.

The volume of a cone is given by the formula,

V = 1/3πr²h

Where V is the volume of cone

r is the base-radius of the cone

and h is the height is perpendicular height of the cone

From the given information,

r = 12 km

h = 24 km

Thus,

V = 1/3 × π × 12² × 24

(Take π = 3.14)

V = 1/3 × 3.14 × 12² × 24

V = 1/3 × 3.14 × 144 × 24

V = 1/3 × 10851.84

V = 3617.28 km³

V ≈ 3617.3 km³ (To the nearest tenth)

Hence, the volume of the given figure is 3617.3 km³

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Answer:

  1/2

Step-by-step explanation:

The "Pythagorean relation" between trig functions can be used to find the sine.

The relation between sine and cosine is the identity ...

  sin(x)² +cos(x)² = 1

This can be solved for sin(x) in terms of cos(x):

  sin(x) = √(1 -cos(x)²)

For the present case, using the given cosine value, we find ...

  sin(x) = √(1 -(√3/2)²) = √(1 -3/4) = √(1/4)

  sin(x) = 1/2

__

Additional comment

The sine and cosine of an angle are the y and x coordinates (respectively) of the corresponding point on the unit circle. The right triangle with these legs will satisfy the Pythagorean theorem with ...

  sin(x)² + cos(x)² = 1 . . . . . . where 1 is the hypotenuse (radius of unit circle)

A calculator can always be used to verify the result.

Substitute [tex]y=\sin^{-1}\left(\frac x4\right)[/tex], so that

[tex]dy = \dfrac{dx}{4\sqrt{1-\left(\frac x4\right)^2}} = \dfrac{dx}{\sqrt{16 - x^2}}[/tex]

Then the integral is

[tex]\displaystyle \int \frac{\left(\sin^{-1}\left(\frac x4\right)\right)^2}{\sqrt{16-x^2}} \,dx = \int y^2 \, dy[/tex]

and by the power rule,

[tex]\displaystyle \int y^2 \, dy = \frac13 y^3 + C[/tex]

so that the original integral is

[tex]\displaystyle \int \frac{\left(\sin^{-1}\left(\frac x4\right)\right)^2}{\sqrt{16-x^2}} \,dx = \boxed{\frac13 \left(\sin^{-1}\left(\frac x4\right)\right)^3 + C}[/tex]

Confidence interval for the mean daily return if it is normally distributed:

⁻x [tex]-z_{\alpha }[/tex](σ/[tex]\sqrt{n}[/tex]) ≤ μ ≤ ⁻x + [tex]z_{\alpha }[/tex] ( σ/[tex]\sqrt{n}[/tex])

Based on the Central Limit Theorem's result that the sampling distribution of the sample means follows an essentially normal distribution, a confidence interval for a population mean is calculated when the population standard deviation is known.

Take into account the standardising equation for the sampling distribution introduced in the Central Limit Theorem discussion:

[tex]z_{1} =[/tex](⁻x - μ₋ₓ) /( σ ⁻x) = (⁻x - μ) /( σ/[tex]\sqrt{n}[/tex])

Notice that µ is substituted for µx− because we know that the expected value of µx− is µ from the Central Limit theorem and σx− is replaced with σn√/, also from the Central Limit Theorem.

In this formula we know X−, σx− and n, the sample size. (In actuality we do not know the population standard deviation, but we do have a point estimate for it, s, from the sample we took. More on this later.) What we do not know is μ or Z1. We can solve for either one of these in terms of the other. Solving for μ in terms of Z1 gives:

μ=X−±Z1 σ/[tex]\sqrt{n}[/tex]

Remembering that the Central Limit Theorem tells us that the distribution of the X¯¯¯'s, the sampling distribution for means, is normal, and that the normal distribution is symmetrical, we can rearrange terms thus:

⁻x [tex]-z_{\alpha }[/tex](σ/[tex]\sqrt{n}[/tex]) ≤ μ ≤ ⁻x + [tex]z_{\alpha }[/tex] ( σ/[tex]\sqrt{n}[/tex])

This is the formula for a confidence interval for the mean of a population.

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By using the digits 0 to 9, the boxes are filled with their respective numerical values to make the chart accurate as shown in the image attached below.

A function can be defined as a mathematical expression which is used to define and represent the relationship that exists between two or more variables.

In Mathematics, there are different types of functions and these include the following;

A logarithm function can be defined as a type of function that represents the inverse of an exponential function. Mathematically, a logarithm function is written as follows:

y = logₐₓ

y = log₁₀10

y = log₁₀10 = 1.  

Therefore, if log₁₀x < 1, then, x < 10.  Also, if log₁₀x > 1, then, x > 10.

For this exercise, you should take note of this deductive points:

In conclusion, by using the digits 0 to 9, the boxes are filled with their respective numerical values to make the chart accurate as shown in the image attached below.

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The complete proof from the figure is:

The given parameter is:

AC ≅ DF; AB ≅ DE; ∠1 ≅ ∠6

Angles 1 and 2, & Angles 5 and 6 are vertical angles

So, we have: ∠1 ≅ ∠2; ∠5 ≅ ∠6 by the definition of vertical angles.

From the figure, triangles ABC and DEF are congruent by the SAS postulate.

So, we have: ΔABC ≅ ΔDEF by the definition of SAS postulate

Angles 3 and 4 are corresponding angles

So, we have: ∠1 ≅ ∠2; ∠5 ≅ ∠6 by the definition of corresponding angles.

Hence, the lines BC and EF are parallel lines

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Answer:

The coordinates of D is (7, 10)

Explanation:

Let the coordinates of endpoint D be (x, y)

Mid-point formula:

[tex]\sf (x_m, y_m) = (\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2})[/tex]

Applying formula:

[tex]\sf (-1, \:4) = (\dfrac{-9+ x}{2},\: \dfrac{-2+y}{2} )[/tex]

Comparing found:

[tex]\sf -1 =\dfrac{-9+ x}{2} \quad and \quad \: 4 = \dfrac{-2+y}{2}[/tex]

[tex]\sf 2(-1)+9 =x \quad and \quad \: 2(4)+2 =y[/tex]

[tex]\sf x = 7 \quad and \quad \: y = 10[/tex]

So, coordinates of D is (7, 10)

The answer is (7, 10).

Remember the midpoint formula : [tex]\boxed {M = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})}[/tex]

Finding the x coordinate of D

Finding the y coordinate at D

The unit rate of Mikayla and brie is 0.16 miles per minutes and 0.25 miles per minutes respectively.

Brie runs faster.

Mikayla

2 miles in 12 1/2 minutes

Unit rate = miles / minutes

= 2 ÷ 12 1/2

= 2 ÷ 25/2

= 2 × 2/25

= 4/25

= 0.16 miles per minutes

Brie

5 miles in 22 1/4 minutes

Unit rate = miles / minutes

= 5 ÷ 22 1/4

= 5 ÷ 89/4

= 5 × 4/89

= 20/89

= 0.25 miles per minutes

Lucy

4 4/5 pounds for $18.50

Unit rate = pounds / price

= 4 4/5 ÷ 18.50

= 24/5 ÷ 18.50

= 24/5 × 1/18.50

= 24/92.50

= $0.26 per pound

Sophie

3 1/2 pounds for $14.75

Unit rate = pounds / price

= 3 1/2 ÷ 14.75

= 7/2 ÷ 14.75

= 7/2 × 1/14.75

= 7/29.5

= $0.24 per pound

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Answer:

c.

Step-by-step explanation:

The two triangles are similar as they have the same angles, though not necessarily the same sides. ASA only applies to congruency.

Answer:

Son: 26 yrs. Dad: 58 yrs.

Step-by-step explanation:

Let s equal to the present age of the son and let d equal to the present age of the dad. We can make two equations based on the information given to us. The first is about the fact that the dad is 32 yrs. older than the son:

d = s+32

the second is about the fact that 10 yrs. ago, he was triple his son's age.

d-10 = 3(s-10)

We can use the distributive property and we have:

d-10 = 3s-30

Then, let's add 10 to each side:

d = 3s-20

We can substitute the first equation into the second, resulting in:

s+32 = 3s-20

Adding 20 to both sides:

s+52 = 3s

2s = 52

s = 26

Now that we have the son's age, the dad's age is 26+32 = 58.

Son's age: 26

Dad's age: 58