How much money would you save if you earned $50,000 a year and maxed your contribution to get the full benefit?

It will cost $1080 to paint all the interior surfaces of the pool.

6. & 7. The triangles have sides of 8, 8, and 10 cm, and an area of 24 square cm. The remaining areas are 18 and 12 square cm, respectively.

Triangle with Sides 8, 8, and 10 cm: First, we must calculate the area of the whole triangle, which has sides of 8, 8, and 10 cm. Using the Heron's formula, we can find the area of a triangle given the lengths of its sides: A = √(s(s - a)(s - b)(s - c)), where s = (a + b + c)/2 is the semi perimeter of the triangle.

A = √(13(13 - 8)(13 - 8)(13 - 10)) = √(13(5)(5)(3)) = √(2925) = 15√(65) ≈ 59.17 square cm. Next, we can calculate the areas of the shaded regions as follows: Area of Shaded Region 1 = Area of Whole Triangle - Area of Small Triangle Area of Shaded Region 1 = 15√(65) - 18 = 15√(65) - 6√(65) = 9√(65) ≈ 27.94 square cm.

Area of Shaded Region 2 = Area of Whole Triangle - Area of Large Triangle Area of Shaded Region 2 = 15√(65) - 24 = 3√(65) ≈ 7.74 square cm.7.

A swimming pool has the dimensions 8 m (length), 6 m (width), and 1.5 m (depth). The total area to be painted is the sum of the areas of the floor, the ceiling, and the walls of the pool. Since the pool is rectangular in shape, we can break it down into six different parts: the top and bottom surfaces, and the four vertical surfaces.

Let's begin by calculating the area of the floor, which has dimensions of 8 m by 6 m:Area of Floor = length x width = 8 x 6 = 48 square meters.

Next, we can calculate the area of the ceiling, which has the same dimensions as the floor: Area of Ceiling = length x width = 48 square meters. Since the pool has a depth of 1.5 m, the total height of the vertical surfaces is 1.5 x 2 = 3 m (since there are two vertical surfaces on each side of the pool).

Thus, the total area of the four vertical surfaces is given by: Area of Walls = perimeter x height = 2(length + width) x height = 2(8 + 6) x 3 = 84 square meters.

Finally, we can calculate the total area to be painted by adding up the area of the floor, ceiling, and walls: Total Area = Area of Floor + Area of Ceiling + Area of Walls Total Area = 48 + 48 + 84 = 180 square meters.

The cost of paint is given as $6 per square meter, so we can calculate the total cost of painting the pool as follows: Cost = Total Area x Cost per Square Meter Cost = 180 x $6 = $1080. Therefore, it will cost $1080 to paint all the interior surfaces of the pool.

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The valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Let's solve each inequality step by step:

1. Two less than a number (x) is no more than 5:

The inequality is given as x - 2 ≤ 5. Adding 2 to both sides, we get x ≤ 7. This means that any number less than or equal to 7 satisfies the inequality.

2. Seven subtracted from 4 times a number (x) is more than 13:

The inequality is given as 4x - 7 > 13. Adding 7 to both sides, we get 4x > 20. Dividing both sides by 4, we obtain x > 5. So any number greater than 5 satisfies the inequality.

3. The sum of two times a number (x) and -2 is at least 8:

The inequality is given as 2x - 2 ≥ 8. Adding 2 to both sides, we get 2x ≥ 10. Dividing both sides by 2, we have x ≥ 5. So any number greater than or equal to 5 satisfies the inequality.

4. Four added to 3 times a number (x) is less than 19:

The inequality is given as 3x + 4 < 19. Subtracting 4 from both sides, we get 3x < 15. Dividing both sides by 3, we obtain x < 5. So any number less than 5 satisfies the inequality.

Based on the above solutions, the valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. These numbers can be represented on a number line graph by marking the appropriate points and shading the corresponding regions.

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Answer:

-5/6

-2/3

Step-by-step explanation:

it's easiest to write the line in the following form:

y=mx+b

where m is the slope

and b is the y intercept

In other words, we want to get y by itself

5x+6y= -4

6y= -4-5x

y=(-4-5x)/6

y=(-5/6)x-4/6

Thus, -5/6 is the slope

and -4/6= -2/3 is the y intercept

Answer:

(x+3)^2 + (y+1)^2 = 16

Step-by-step explanation:

The equation of a circle is (x – h)^2 + (y – k)^2 = r^2, where h is the x value of the center, k is the y value of the center, and r is the radius.

We can see from the picture that the radius is at about (-3, -1) and the radius is about 4, so we can plug those in:
(x – (-3))^2 + (y – (-1))^2 = 4^2

Simplify:
(x+3)^2 + (y+1)^2 = 16

Answer:

Equation of circle:[tex](x + 3)^2 + (y + 1)^2 = 16[/tex]

Step-by-step explanation:

Given:

Center of the circle = (-3, -1)

Point on the circle = (1, -1)

In order to find the radius of the circle, we can use the distance formula.

distance =[tex] \boxed{\bold{\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}}}[/tex]

where:

In this case, the distance formula becomes:

radius = [tex]\sqrt{(-3 - 1)^2 + ((-1) - (-1))^2}= \sqrt{16}=4[/tex]

Therefore, the radius of the circle is 4 units.

Now that we know the radius of the circle, we can find the equation of the circle using the following formula:

[tex]\boxed{\bold{(x - h)^2 + (y - k)^2 = r^2}}[/tex]

where:

In this case, the equation of the circle becomes:

=[tex](x + 3)^2 + (y + 1)^2 = 4^2[/tex]

=[tex](x + 3)^2 + (y + 1)^2 = 16[/tex]

This is the equation of the circle.

Answer/ Step-by-step explanation:

---

Proportional relationships are relationships between two variables where their ratios are equivalent.

---

Lets take a look at example one attached image:

The constant of proportionality is the value of y when x = 1 in a proportional relationship.

On which line does y = 1/2 when x = 1?

Only line C has a y- value less than 1 when x = 1, but how can we be sure that the constant of proportionality is exactly 1/2?

If the constant of proportionality of a proportional relationship is 1/2 then:

y = 1/2x (like rise over run (slope), rise up 1 unit, and run right 2 units)

We can try the point (2, 1), and plug in the values into the equation, where x = 2, y = 1.

(1) = 1/2(2)

1 = 1

Line C has a constant proportionality of 1/2 between x and y.

--------

Example 2, 3, and 4 is a worked out example on khan academy (attached images)

The equation of the line that passes through the point (-3, 7) and has a slope of -5/3 is y - 7 = (-5/3)(x + 3).

We are given the point (-3, 7) and the slope of the line as -5/3.The slope-intercept form of a line is y = mx + b where m is the slope and b is the y-intercept.

To obtain the equation of the line, we need to substitute the values of slope and point in the slope-intercept form and solve for b.(7) = (-5/3)(-3) + b 21/3 = b.

Now we have the value of b, and we can substitute the values of m and b in the slope-intercept form.y = (-5/3)x + 21/3 is the equation of the line in slope-intercept form.

To obtain the equation in the standard form Ax + By = C, we multiply each term by 3.3y = -5x + 7Add 5x to both sides5x + 3y = 7.

This is the equation of the line in standard form.

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The roster method of the set A' = {14, 15, 20}

The roster method is defined as a way to show the elements of a set by listing the elements inside of brackets.

A typical example of the roster method is to write the set of numbers from 1 to 5 as {1, 2, 3, 4, 5}.

Therefore,

U = universal sets = {14, 15, 16, 17, 18, 19, 20}

A = {16, 17, 18, 19}

Therefore, let's find A' as follows:

A' is the value not in A. Hence, in roster method.

A' = {14, 15, 20}

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Answer: y = (-28)x + 14

Step-by-step explanation:

The equation for the line tangent at point (a, b) can be found using the formula y = mx + c, where m is the slope of the line and c is the y-intercept. At point (6, 14), we need to find the value of m.

We start by finding the derivative of the original function g(x). Its derivative is given by:

dg(x)/dx=7/(x^2-3x+1)

Then, substitute x = 6 into the derivative expression to obtain:

dg(6)/dx = d/dx [7 * ln|x-3| ] evaluated at x = 6 = 7/3

Next, evaluate the original function g(x) at x = 6 to get g(6) = 7 * ln |6 - 3| / (6 - 3) = 7 * ln 3.

Since we know the coordinates of the point of tangency (6, 14), we can substitute them into the general form of the linear equation y = mx + c:

14 = 7 * 6 + c

14 = 42 + c

c = -28

The final equation of the line tangent at point (6, 14) is therefore:

y = (-28)x + 14

The angle measure that belongs in the green box is given as follows:

70.67º.

We consider a triangle with side lengths and angles related as follows, as is the case for this problem:

Then the lengths and the sines of the angles are related as follows:

sin(A)/a = sin(B)/b = sin(C)/c.

The relation for this problem is given as follows:

sin(x)/12  = sin(76º)/12.34

Applying cross multiplication, the missing angle measure is given as follows:

sin(x) = 12 x sine of 76 degrees/12.34

sin(x) = 0.9436.

x = arcsin(0.9436)

x = 70.67º.

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Answer: 1000

Step-by-step explanation:

If one person can pick up a plate every 10 seconds, then the rate of picking up plates can be represented as 1 plate/10 seconds.

When two people are picking up plates, the rate doubles, so it becomes 2 plates/10 seconds or 1 plate/5 seconds.

To find how long it will take for 200 people to pick up plates, we need to use the formula:

time = amount ÷ rate

Plugging in the values, we get:

time = 200 plates ÷ (1 plate/5 seconds)

time = 200 plates × 5 seconds/plate

time = 1000 seconds

Therefore, it will take 1000 seconds for 200 people to pick up plates at a faster rate.

I hope that this answer has helped you!

Answer: 33.7

Step-by-step explanation:

Let the height of the house be h, and the distance from the foot of the rock to the house be x.

From the given information, we have the following diagram:

```

*

/ \

/ \

/ 63° \

/ \

/θ \

A ----------------------- B

70m x

Angle A = 60° (complementary to the angle of elevation of the foot of the house)

Angle B = 63° (angle of elevation of the top of the house)

Using trigonometry, we have:

tan(63°) = h/x ----(1) (for triangle AOB)

tan(60°) = h/(x + 70) ----(2) (for triangle ABD)

Solving equations (1) and (2) simultaneously, we get:

h = (70 tan(63°) - 70 tan(60°)) meters

h ≈ 33.7 meters (rounded to one decimal place)

Therefore, the height of the house is approximately 33.7 meters.

The solution above is graphed correctly in the last option choice is x > -6

We have been given the equation 2x + 3> -9

In order to graph the solution, we must find the value of x

2x + 3 > -9

Subtract three from both sides

-9 - 3 = -12

2x > -12

Divide both sides by 2

x > -6

Examine the inequality symbol to figure out how to graph the solution. If the sign is "greater than," graph the line to the left. If it was "less than," you would draw a straight line.

We have the "greater than" symbol in our issue, which implies we will graph our line to the right, and since we start our line at -6, we know the last choice is the correct solution.

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The following question may be like this:

Which graph shows the solution set for 2x+3>-9.

The members of the given set in this problem are given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

The union and intersection sets of multiple sets are defined as follows:

The intersection of the sets Y and Z for this problem is given as follows:

Y ∩ Z = {0, 6, 23, 26}

(which are the elements that belong to both of the sets).

The union of the above set with the set X is given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

(elements that belong to at least one of the sets).

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Answer:

The product of each pair of complex conjugates is:

(3 + 8i)(3 – 8i) = (9 - 8^2) + i(9 + 8^2) = (-81) + i(99) = (-81) + i(11) = (-81) + 11i

And

(4 + 5i)(4 – 5i) = (16 - 25) + i(16 + 25) = (-9) + i(41) = (-9) + 41i

So, the products are:

(-81) + 11i

and

(-9) + 41i

Answer:

73

41

Step-by-step explanation:

Answer: x = 3 and x = -6

Step-by-step explanation:

the first step is to square root both sides of the equation to get rid of the exponent (2) on the left side of the equation, however lets break it down:

[tex](2x + 3)^2 = 81\\\\[/tex]

81 can be rewritten as 9^2

[tex](2x + 3)^2= 9^2[/tex]

and now lets square root both sides:

[tex]\sqrt{(2x+3)^2} =\sqrt{9^2}[/tex]

The squares (the exponent 2) cancels out with the square root:

2x + 3 = +/- 9

now lets isolate x by subtracting 3 from both sides:

2x + 3 = +/- 9

       -3  -3

2x = -3 +/- 9

2x = -3 + 9

2x = 6

2x = -3 - 9

2x = 12

And after simplifying, you can divide two on both sides:

2x = 6

/2    /2

x = 3

2x = -12

/2    /2

x = -6

x = 3 and x = -6

As the given equation is (36)ˣ⁺²  = 6 , then the solution for the given equation is x ≈ -1.54.

The given equation in the question is (36)ˣ⁺²  = 6

according to the given equation , the goal is to solve the equation for x.

For solving this equation ,

we'll have to use logarithmic operations which is as follows:  

log36(6) = x+2.

We will be further using a calculator to determine the logarithmic value: log36(6) ≈ 0.4607.

Now, By subtracting 2 from both sides, we will get the value of x as:

x ≈ -1.54.

When we round off this value to the nearest 100th, we will get  -1.54.

Thus , the solution for the given equation is x ≈ -1.54.

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Answer: The two solutions are (2, 4) and (-1, 1).

Step-by-step explanation:

The given is a system of equations, y = x + 2, and, y = x^2

A system of equations comprises two or more equations and seeks common solutions to the equations.

To solve you can use substitution. And by doing so you can replace y of one equation with what the other equation equals:

y = x + 2

y = x^2

->   x + 2 = x^2

Now lets get all variables and constant to one side of the equation, set it equal to zero.

x + 2 = x^2

-x -2           -x -2

x^2 -x -2 = 0

Lets factor to find the solution

Factoring is used to simplify an algebraic expression by finding the greatest common factors that are shared by the terms in the expression.

We can factor, x^2 -x -2 = 0, into:

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

-------

SIDE NOTE:

If confuse on factoring please see attached image.

------

with (x - 2)(x + 1) = 0 we can find the solution by setting each component in the parentheses to 0:

x - 2 = 0

x + 1 = 0

Now you must solve for x.

x - 2 = 0

+2      +2

x = 2

x + 1 = 0

  -1      -1

x = -1

x = 2 and x = -1

However, we are not done yet, we need to find what y is, and by doing so we can plug in the x values we got to find the corresponding y value to create points (coordinates).

-

When x = 2

y = x + 2

y = (2) + 2

y = 4

(2, 4)

-

When x = -1

y = x + 2

y = -1 + 2

y = 1

(-1, 1)

-

The two solutions are (2, 4) and (-1, 1).

The truth values of the statements can be determined as follows:

Statement p: "A square has 4 sides."

This statement is true. A square is a polygon with four equal sides and four equal angles, so it has 4 sides.

Statement q: "A triangle has 5 sides."

This statement is false. A triangle is a polygon with three sides and three angles, not five. It is a fundamental geometric shape with specific characteristics, and one of those characteristics is having three sides.Therefore, the truth values of the statements are:

p is true.

q is false.

It's important to note that truth values are based on established definitions and properties of geometric shapes. The statement p aligns with the definition of a square, while the statement q contradicts the definition of a triangle.

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Answer: 50$

Step-by-step explanation:

To calculate the amount the college student can spend on fast food in December, we need to find the total amount he can spend in 5 months, given that he wants to spend an average of $50 per month.

Let's start by finding the total amount he can spend in 5 months:

Total amount = 5 x $50 = $250

Now, we can subtract the amount he spent in the other 4 months from the total amount to find out how much he can spend in December:

The amount he can spend in December = Total amount - Amount spent in 4 months

Amount spent in 4 months = $35 + $90 + $15 + $60 = $200

Amount he can spend in December = $250 - $200 = $50

Therefore, the college student can spend $50 on fast food in December to meet his goal of spending an average of $50 per month on fast food for the remaining 5 months of the year.

Step-by-step explanation:

The circumference of a circle can be found using the formula:

C = πd

where C represents the circumference and d represents the diameter of the circle.

Given that the diameter of the circle is 4 feet, we can substitute this value into the formula:

C = π * 4

Therefore, the circumference of the circle is 4π feet.

The values of p and q are 1 and 3, respectively.

Given the equation x^2-4x + 1 = (x-p)- q, we can compare the coefficients of the corresponding terms on both sides of the equation.

We compared the coefficients of the x^2 terms on both sides of the equation. The coefficient of the x^2 term on the left-hand side is 1, and the coefficient of the x^2 term on the right-hand side is 1. This means that the two terms are equal, and therefore p = 1.

We compared the coefficients of the x terms on both sides of the equation. The coefficient of the x term on the left-hand side is -4, and the coefficient of the x term on the right-hand side is -1. This means that the two terms are equal, and therefore q = 3.

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Avoidable costs are costs that can be eliminated if the activity that caused them is discontinued. The examples are listed.

Direct materials: Direct materials are the materials that go into a product and can be easily traced to it. For example, the wood used to make a table is a direct material. If a company decides to stop making tables, it can avoid the cost of buying wood.

Direct labor: Direct labor is the labor that is directly involved in making a product. For example, the wages paid to the workers who assemble a car are direct labor. If a company decides to stop making cars, it can avoid the cost of paying direct labor.

Variable overhead: Variable overhead is the overhead costs that vary with the number of units produced. For example, the cost of electricity used to power a factory is a variable overhead cost. If a company decides to stop producing a product, it can avoid the variable overhead costs associated with that product.

Sunk costs: Sunk costs are costs that have already been incurred and cannot be recovered. For example, the cost of research and development for a new product is a sunk cost. If the company decides not to launch the product, it cannot recover the cost of research and development.

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The statement that is true about quadrilateral QRST is: d. Side TQ has a length of sq rt of 17 units

A quadrilateral is a polygon with four sides and four vertices. The term "quadrilateral" is derived from the Latin words "quadri" meaning "four" and "latus" meaning "side."

Quadrilaterals come in various shapes and sizes, and they can have different angles and side lengths. Some common examples of quadrilaterals include rectangles, squares, parallelograms, trapezoids, and rhombuses.

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The complete question is:

Which statement is true about quadrilateral QRST?

a. Side QR has a length of 5 units.

b. Side RS has a length of sq rt of 40 units.

c. Side ST has a length of 6 units.

d. Side TQ has a length of sq rt of 17 units

Using concentration-volume relationship, 30 ml of potassium chloride should be added to the IV solution.

To calculate the amount of potassium chloride to be added to the IV, we need to consider the concentration of the potassium chloride ampoules and the desired final concentration in the IV solution.

Given:

Physician's order: Add 60mEq of potassium chloride to 1000ml D5W

Potassium chloride concentration in ampoules: 40 mEq per 20 ml

Desired final concentration: Not specified

To determine the amount of potassium chloride to be added, we can set up a proportion based on the mEq (milliequivalents) of potassium chloride:

This is done using concentration-volume relationship

40 mEq / 20 ml = 60 mEq / x ml

Cross-multiplying, we have:

40 mEq * x ml = 60 mEq * 20 ml

Simplifying:

40x = 1200

Dividing both sides by 40:

x = 30 ml

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The radius of the semicircle protractor is approximately 4.693 cm.

Given,Perimeter of a semicircle protractor = 14.8 cm.

To find:The radius of a semicircle protractor.Solution:We know that the perimeter of a semicircle protractor is the sum of the straight edge of a protractor and half of the circumference of the circle whose radius is the radius of the protractor.

Circumference of a circle = 2πrWhere, r is the radius of the circle.If the radius of the semicircle protractor is r, then Perimeter of a semicircle protractor = r + πr [∵ half of the circumference of a circle =[tex](1/2) × 2πr = πr]14.8 = r + πr14.8 = r(1 + π) r = 14.8 / (1 + π)r ≈ 4.693[/tex] cm.

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The calculated value of x in the figure is 12

From the question, we have the following parameters that can be used in our computation:

The figure

The scale is given as

A : B = 8 : 9

So, we have

48 : 2x + 30 = 8 : 9

Next, we have

(2x + 30)/48 = 9/8

This gives

2x + 30 = 48 * 9/8

2x + 30 = 54

This gives

2x = 24

So, we have

x = 12

Hence, the value of x in the figure is 12

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The probability is 5/8 that the second number chosen will exceed the first number by a distance greater than 1/4 unit on the number line when two numbers are randomly chosen between 0 and 1.

To determine the probability that the second number chosen will exceed the first number by a distance greater than 1/4 unit on the number line, we need to consider the possible range of values for both numbers.

Since the two numbers are chosen randomly between 0 and 1, we can visualize this as a unit interval on the number line, where 0 represents the left endpoint and 1 represents the right endpoint.

Let's analyze the scenario where the first number is chosen and labeled as x. The probability that the second number chosen will exceed x by a distance greater than 1/4 unit can be represented by the shaded area on the number line.

To calculate this probability, we need to determine the length of the interval where the second number can be chosen, given that it exceeds x by more than 1/4 unit.

If the first number, x, is chosen between 0 and 3/4 (i.e., x < 3/4), the second number can be chosen in the range (x + 1/4, 1]. The length of this interval is 1 - (x + 1/4) = 3/4 - x.

If the first number, x, is chosen between 3/4 and 1 (i.e., 3/4 ≤ x < 1), the second number can be chosen in the range (3/4, 1]. The length of this interval is 1 - 3/4 = 1/4.

Since the probabilities of choosing a number within each interval are equally likely, we need to calculate the weighted average of these probabilities based on the lengths of the intervals.

The probability of choosing a number between 0 and 3/4 is (3/4 - 0) = 3/4, and the probability of choosing a number between 3/4 and 1 is (1/4 - 0) = 1/4.

Therefore, the overall probability is calculated as:

P = (3/4) * (3/4) + (1/4) * (1/4) = 9/16 + 1/16 = 10/16 = 5/8.

So, the probability that the second number chosen will exceed the first number by a distance greater than 1/4 unit on the number line is 5/8.

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Answer:

.6

Step-by-step explanation:

The scaled triangle will be larger than the initial size by a factor 2.

The scaled square will be smaller than the initial side by a factor 4

Dilation refers to a transformation that changes the size of a geometric figure without altering its shape.

Dilation involves scaling an object by a certain factor, that might result in  enlarging or reducing its dimensions uniformly in all directions.

Based on the given diagram, the new length and size of the object is calculated as follows;

For the triangle, (measure the length with ruler)

For the square; (measure the length with ruler)

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