Part A: Sydney made $18. 50 selling lemonade, by the cup, at her yard sale. She sold each cup for $0. 50 and received a $3 tip from a neighbor. Write an equation to represent this situation. (4 points) Part B: Daria made a profit of $21. 00 selling lemonade. She sold her lemonade for $0. 75 per cup, received a tip of $3 from a neighbor, but also had to buy each plastic cup she used for $0. 10 per cup. Write an equation to represent this situation. (4 points) Part C: Explain how the equations from Part A and Part B differ. (2 points

The present ages of Raj and his daughter are respectively: 42 years and 12 years.

Present Age of Raj =y years

Present Age of daughter =x years

According to Question :

7 Years ago,

y − 7 = 7(x − 7)

⇒ 7x − y − 42 = 0.............(1)

And 3 Years from now

y + 3 = 3(x + 3)

⇒ 3x − y + 6 = 0............(2)

From eq (1) and eq (2)

Subtract eq 2 from eq 1 to get:

7x − 3x − y + y − 42 − 6 = 0

⇒ 4x = 48

⇒ x = 12

Putting x = 12 in Equation (2). we get,

(3 × 12) − y + 6 = 0

⇒ y = 42

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To find the average amount that Sarah saved per month over the last six months, we need to add up the total amount saved and divide by the number of months.

Total amount saved = $135.00 + $144.00 + $104.00 + $80.00 + $90.00 + $160.00 = $713.00

Number of months = 6

Average amount saved per month = Total amount saved / Number of months = $713.00 / 6 = $118.83

Therefore, the correct answer is d. $118.83.

It is important to note that when working with numbers and calculations, accuracy is crucial. In this case, rounding off the answer to the nearest cent would result in a different answer.

Additionally, checking the calculations multiple times to ensure accuracy is always recommended.

Overall, tracking and analyzing expenses and savings is important for financial planning and achieving financial goals. By keeping track of how much she saved each month,

Sarah can make informed decisions about her spending and saving habits and adjust accordingly to reach her vacation savings goal.

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The number of bricks that will be required to build the staircase is: 30 bricks

An arithmetic sequence is defined as one where you get the next term by adding a constant, called the common difference, to the previous term. A lot of formulas come from this simple fact. and they allow us to solve for any term in the sequence and even the sum of the first few terms.

The formula for the nth term of an arithmetic sequence is:

aₙ = a₁ + (n - 1)d

where:

a₁ is first term

d is common difference

n is nth term

We are given:

a₁ = 200

d = -4

aₙ = 84

Thus:

84 = 200 + (n - 1)(-4)

84 - 200 = -4n + 4

-4n = -120

n = 30

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Kate's mistake was in Step 2: "Divide the volume of Wesley's tank by 2." Since she wants a tank with half the volume of Wesley's tank, she should have divided the volume of Wesley's tank by 2, not the radius.

The correct approach would be to use the formula for the volume of a sphere:

V = (4/3)πr^3

Since Wesley's tank has a diameter of 30 cm, its radius is 15 cm. Substituting this value into the formula gives:

V = (4/3)π(15)^3

V ≈ 14,137 cm^3

To find the radius of the smaller tank with half the volume, Kate should have solved for the radius using the formula for the volume of a sphere:

(4/3)πr^3 = (1/2)(4/3)π(15)^3

Simplifying this equation gives:

r^3 = (1/2)(15)^3

r ≈ 10.8 cm

Therefore, the approximate diameter of the smaller tank should be 21.6 cm (twice the radius), not 10.8 cm.

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In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.

When Amelie divides the total number of muffins (385) by the number of muffins per box (9), she obtains a quotient of 42 and a remainder of 7. The quotient represents the number of complete boxes that Amelie can fill with 9 muffins each, while the remainder represents the number of muffins that cannot be put into a full box.

In other words, the quotient tells us how many full boxes Amelie can pack, and the remainder tells us how many muffins are left over after packing all the full boxes. In this case, Amelie can pack 42 full boxes, each with 9 muffins, which totals to 378 muffins. The remaining 7 muffins cannot fill a full box, and they are left over after all the full boxes are packed.

The "R 7" notation is commonly used to indicate the remainder in long division problems. The letter "R" stands for "remainder," and the number following it (in this case, 7) represents the actual remainder. The remainder is important because it indicates how many items are left over after dividing them into equal-sized groups.

In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.

Overall, interpreting the "R 7" in this problem helps us understand how many full boxes of muffins Amelie can pack and how many muffins are left over after packing the full boxes.

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The measure of angle x of the ninja star is given by x = 56°

Given data ,

Let the ninja star be represented as an isosceles triangle

Now , it is constructed using four congruent isosceles triangles that are placed along the sides of a square

And , the vertex angle of the triangle is x

In an isosceles triangle, the two equal sides are called legs, and the remaining side is called the base. The angle opposite the base is called the vertex angle

So , the measure of x = 180° - ( 62° + 62° )

On simplifying , we get

x = 56°

Hence , the angle of triangle is x = 56°

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The current number of classrooms is 24, and the number of students per class is 70 if there were a total of 1200 students.

Let us assume that the number of classes = x

Number of students per class = 1200/x

Number of classrooms planned = x - 4

Number of students planned per class = 1200/ x+10

Total number of students = 1200

By using the above data, the equations will be written as:

(1200 / x-4) = (1200/x) +10

By multiplying the equation 2 we get:

1200x = 1200x + [tex]x^{2}[/tex] - 4800 - 40x

[tex]x^{2}[/tex] - 480- 4x = 0

(x-24) (x+20) = 0

x = 24

Number of rooms built = x =24

Number of students per class = (1200/24-10) = 60 students

Therefore, we can conclude that the current number of classrooms is 24, and the number of students per class is 60 + 10 =70.

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Mark and Diane's property tax is $3,231.80.

To calculate the property tax for Mark and Diane, we need to multiply the assessed value of their home by the tax rate.

First, we need to convert the tax rate from mills to a decimal. One mill is equal to one-tenth of one cent or 0.001 dollars. Therefore, 43.7 mills is equal to 0.0437 dollars.

Next, we can calculate the property tax as follows:

Property tax = Assessed value × Tax rate

Property tax = $74,000 × 0.0437

Property tax = $3,231.80

Therefore, Mark and Diane's property tax is $3,231.80.

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The probability of drawing a yellow marble on the next draw is 13/48, which is option A, 13%.

The probability of drawing a yellow marble in the next draw depends on whether the drawing process is with or without replacement.

If the drawing process is with replacement, meaning that the marble is put back into the bag after each draw, then the probability of drawing a yellow marble remains the same at 13/48.

If the drawing process is without replacement, meaning that the marble is not put back into the bag after each draw, then the probability of drawing a yellow marble changes. After 48 draws, there are 35 white marbles and 13 yellow marbles left in the bag.

Therefore, the correct answer is A) 13%.

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71.38 in word form is Seventy-one and thirty-eight hundredths.

What is the decimal number?

The accepted method for representing both integer and non-integer numbers is the decimal numeral system. Decimal notation is the term used to describe the method of representing numbers in the decimal system.

A number is a numerical unit of measurement and labeling in mathematics. The natural numbers 1, 2, 3, 4, and so on are the first examples. Number words are a linguistic way to express numbers.

To write decimals in word form, you can read the whole number part, write "and," and then read the decimal part according to place value.

For example: 0.25 can be written as "zero and twenty-five hundredths"

Hence, the correct answer is 'D. Seventy-one and thirty-eight hundredths'.

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The value of f(-2)  is 15.

Hence option D is correct.

The given expression is

[tex]\int\limits {f(x)} \, dx[/tex] = (1/3)x³ - 2x² +3x - 5

To find expression of f(x)

Differentiate it with respect to x

f(x) = x² - 4x + 3

Now put x = -2

f(-2) = (-2)² - 4(-2) + 3

       = 4 + 8 + 3

       = 15

Hence,

f(-2)  = 15

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The fair return for Castle Black Camping (CBC) is 20%.

This is calculated by subtracting the risk-free rate (2%) from the market's expected return (11%), and then multiplying the result by 2 (since CBC is twice as risky as the average stock).

To form a portfolio that is half Harlan County Safety Co. (HCSC) and half CBC, we need to calculate the portfolio's beta. This is done by multiplying each stock's beta by its weight in the portfolio, and then adding the results. In this case, the portfolio's beta would be 0.6 (1.2 x 0.5 + 2 x 0.5).

The fair return for the portfolio is then calculated by adding the risk-free rate to the product of the portfolio's beta and the market's expected return. This gives us a fair return of 16.4% for the HCSC and CBC portfolio.

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Answer:

Principal = $21,592.31

Step-by-step explanation:

The formula for continuous compound interest is

[tex]A = Pe^r^t[/tex], where A is the amount (aka investment worth), r is the interest rate, and t is the time in years (the number e simply shows us that we're dealing with continuous compound interest)

Since we're already given that have A = $58,226.31, r = 0.062 (we must convert the percentage to a decimal by simply moving the decimal two places to the left, which is the same as dividing by 100), and t = 16 years, we can simply solve for P:

[tex]58226.31=Pe^(^0^.^0^6^2^*^1^6^)\\58226.31=Pe^0^.^9^9^2\\58226.31/(e^0^.^9^9^2)=P\\21592.31176=P\\21592.31=P[/tex]

To construct a confidence interval for the average bonus that all employees working for financial companies in New York City received last year, we need to know the sample size and the level of confidence. Without this information, we cannot calculate the confidence interval. Please provide the sample size and the level of confidence to proceed with the solution.

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The pairs of cities that will not put Jenny back under budget if she drops them are:

a. Moscow and Chelyabinsk

b. Izhevsk and Nizhny Novgorod

To determine which pair of cities Jenny can drop without going over budget, we need to find the total cost of each pair and see if it exceeds her budget of rub 33,000.

a. Moscow and Chelyabinsk:

Total cost = 5,485 + 5,217 = 10,702 rubles

This pair of cities is over budget, so Jenny cannot drop any other cities if she wants to visit both Moscow and Chelyabinsk.

b. Izhevsk and Nizhny Novgorod:

Total cost = 4,721 + 6,920 = 11,641 rubles

This pair of cities is over budget, so Jenny cannot drop any other cities if she wants to visit both Izhevsk and Nizhny Novgorod.

c. Novosibirsk and Saint Petersburg:

Total cost = 4,870 + 5,960 = 10,830 rubles

This pair of cities is under budget, so Jenny can drop this pair without going over budget.

d. Yaroslavl and Tolyatti:

Total cost = 4,901 + 5,598 = 10,499 rubles

This pair of cities is under budget, so Jenny can drop this pair without going over budget.

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Answer: 90 feet squared

Step-by-step explanation:

The formula for finding the area of a triangle is

(bxh)/2

In this problem, you need to convert 3 yards to feet. The conversion is as follows:

1 yard = 3 feet

Therefore, 3 yards = 9 feet.

You can then plug everything in and solve as follows:

20x9= 180

Then divide 180 by 2.

180/2=90

Answer:[tex]90ft^{2}[/tex]

Step-by-step explanation:

Area of a triangle: A=(bh)/2

b = 20 ft

h = 3 yds = 9ft

Plug in

A=[(20ft)(9ft)]/2

A=180ft/2

A=90ft^2

Answer:

--------------------

First, plot the points and connect together in the given order.

See attached.

As we can see all sides are of different length but two sides seem to be parallel.

Let's find the slopes of segments BC and AD and compare:

As se see the slopes are same, so BC and AD are parallel.

Since we have a pair of parallel sides, the quadrilateral is a trapezoid.

This circle with center C will be circumscribed around triangle △DEF, touching each vertex. The translated and dilated ⊙O will coincide with ⊙P, proving that they are similar.

1. To construct a circle with center C circumscribed about triangle △DEF, first find the perpendicular bisectors of all three sides of the triangle. Next, locate the point where the three perpendicular bisectors intersect, which will be the center C of the circumscribed circle. Finally, draw the circle using the distance from point C to any vertex of the triangle as the radius.

2. To prove that ⊙O and ⊙P are similar using similarity transformations on ⊙O, follow these steps:
  a. Translate ⊙O by the vector <-2+12, 7-(-1)> or <10, 8> so that its center coincides with the center of ⊙P. The translated ⊙O will have the center (10, 15).
  b. Since the ratio of the radii of ⊙O and ⊙P is 5:12, perform a dilation of ⊙O with a scale factor of 12/5 and center (10, 15). The dilated ⊙O will have a radius of 12, which is equal to the radius of ⊙P.
 

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The coordinate of the image point is (9,1).

There are eight types of rules for the transformation of a point. When the  takes place across a line, then the point (x,y) is changed to the point (y,x).

Given that the rule for the transformation of a point P(1,1) is [tex]R_{x=5} (P)[/tex], which defines the reflection of a point about a line, that is parallel to the y-axis. The line [tex]x=5[/tex] is like a mirror. So, the distance between the line and the image point is equal to the distance between the line and the original point.

Using the point-line distance formula, the distance between the line [tex]x=5[/tex] and a point (1,1) is given by [tex]|5-1|=4[/tex].

Similarly, by the above statement, the distance between the line [tex]x=5[/tex] and the image point will also be 4.

Therefore, the coordinate of the image point is (9,1).

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The complete question is -

If P = (1,1), then find the reflection [tex]R_{x=5} (P)[/tex].

The point on the number line that represents the probability of rolling an even number and selecting a green marble is 3/8, which is between 0.3 and 0.4 on the number line.

Will she land on an even number and then selects agreen marble?

The probability of rolling an even number is 3/6, which can be simplified to 1/2, because there are three even numbers (2, 4, and 6) out of six possible outcomes.

The probability of selecting a green marble from the bag is 3/4, because there are three green marbles out of four total marbles in the bag.

To calculate the probability of both events happening together (rolling an even number and selecting a green marble), you multiply the probabilities of each event:

P(even number and green marble) = P(even number) x P(green marble)

P(even number and green marble) = (1/2) x (3/4)

P(even number and green marble) = 3/8

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10 stereos must the salesperson sell to make that same amount of money under both plans.

Assume that in order for the salesperson to earn the same amount of money under both schemes, x stereos must be sold.

The salesperson will be paid a salary of $250 + $25 for each stereo sold under the first strategy. So, these are the total earnings:

First plan's total profits are $250 plus $25.

Instead of receiving a salary under the second proposal, the salesperson would be compensated with a $50 commission for each stereo sold. So, these are the total earnings:

Total income under the second plan equals $50x

We can set the two equations for total earnings to be equal in order to determine the value of x:

$250 + $25x = $50x

When we simplify the equation, we obtain:

$250 = $25x

x = 10

The sales person must thus sell 10 stereos in order to.

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Example problem:

Solution:

Using the formula for exponential decay:

[tex]\sf\qquad\dashrightarrow N(t) = N0 * e^{(-kt)}[/tex]

where:

Since the half-life is 10 years, we know that:

[tex]\sf\qquad\dashrightarrow k = \dfrac{\ln(0.5)}{10} = -0.0693[/tex]

Substituting the given values, we get:

[tex]\sf:\implies N(30) = 100 * e^{(-0.0693 * 30)}[/tex]

[tex]\sf:\implies N(30) = 100 * e^{(-2.079)}[/tex]

[tex]\sf:\implies N(30) = 100 * 0.126[/tex]

[tex]\sf:\implies \boxed{\bold{\:\:N(30) = 12.6\: grams\:\:}}\:\:\:\green{\checkmark}[/tex]

Therefore, after 30 years, only 12.6 grams of the radioactive substance will be left.

The total number of customers that the waiter had would be = 30 customers.

The total number of tables the waiter had = 5 tables

The total number of women at each table = 3

The total number of men at each table = 3

The total number of people one each table = 6

Therefore the total number of customers that the waiter attended to would be = 5×6 = 30

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(a) Discount amount is $6. (b) Maturity value is $2,200. (c) Current yield is 0.27%.

Here, we have,

A discount is a reduction in the regular selling price of a good or service, either in terms of money or as a percentage.

A product may, for instance, be discounted by $10 from its list price or by 10% from its list price. A sales discount is a lower price that a company offers on a good or service.

Find out how to add discounts to invoices. A sales discount, usually referred to simply as a "discount," offers clients of a business a lower price on one or more of the goods or services being provided.

(a) Discount amount

= Maturity (face) value - Purchase price

=$2,200 - $2,194

=$6

(b) Maturity value

=Face value

=$2,200

(c) Current yield

=Discount amount / Purchase price

=$6 / $2,194

=0.0027, or 0.27%

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The common ratio for the geometric series 1, 1/2, 1/4, 1/8, ... is 1/2 and the sum of series is 2 which is finite.

To determine whether the series has an infinite sum, we can use the formula for the sum of an infinite geometric series, which is:

S = a/(1-r),

where a is the first term and r is the common ratio.

In this case, a = 1 and r = 1/2, so

S = 1/(1 - 1/2) = 2.

Since the value of S is finite and not infinite, we can conclude that the given geometric series has a finite sum of 2.

The common ratio of a geometric series is the ratio between consecutive terms. For example, in the series 1, 2, 4, 8, 16, ..., the common ratio is 2 because each term is obtained by multiplying the previous term by 2.

To determine whether a geometric series has an infinite sum, we can use the formula for the sum of an infinite geometric series, which is S = a/(1-r), where a is the first term and r is the common ratio.

If the value of r is between -1 and 1 (excluding -1), then the series has a finite sum. If the value of r is greater than 1 or less than -1, then the series has an infinite sum.

In the given series 1, 1/2, 1/4, 1/8, ..., the common ratio is 1/2. To find the sum of the series, we can use the formula S = a/(1-r) with a=1 and r=1/2, which gives S=2. Since the value of S is finite and not infinite, we can conclude that the given geometric series has a finite sum of 2.

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complete question:

Determine the common ratio for each of the following geometric series and determine which ones have an infinite sum 1,1/2,1/4,1/8....

The optimal rental price to maximize revenue is $35, the same as two summers ago.

To determine the optimal canoe rental price to maximize revenue, we can use the concept of price elasticity of demand, which measures the responsiveness of demand to a change in price.

When the price of a product decreases, consumers tend to buy more of it, but the increase in demand may not be proportional to the decrease in price. The price elasticity of demand can help us estimate the percentage change in demand for a given percentage change in price.

In this case, we can use the data from the previous two summers to estimate the price elasticity of demand for canoe rentals. From the data provided, we know that a $5 decrease in price led to an increase of 25 canoes rented.

This means that the price elasticity of demand is approximately -5 (25/5). In other words, for every 1% decrease in price, we can expect a 5% increase in demand.

To determine the optimal rental price, we need to find the point where the marginal revenue from renting an additional canoe is equal to the marginal cost of renting it out. Assuming that the marginal cost of renting out an additional canoe is constant, we can use the price elasticity of demand to estimate the change in revenue due to a change in price.

If we increase the rental price by $1, we can expect to lose 5% of customers (assuming the same elasticity as last summer). This means that for every $1 increase in price, we will lose 7.5 (150*5%) customers. On the other hand, we will gain $35 in revenue for each of the remaining 142.5 canoes rented, resulting in a total revenue of $4,987.5.

If we decrease the rental price by $1, we can expect to gain 5% more customers, resulting in 157.5 canoes rented. However, we will also lose $30 in revenue for each of the 150 original customers who decide to rent at the lower price.

This means that for every $1 decrease in price, we will gain 7.5 customers but lose $4,500 in revenue. The total revenue at a rental price of $34 will be $4,827.5.

This price will result in the same number of customers as two summers ago but with a slightly higher revenue due to inflation.

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Answer:

3.966m

Step-by-step explanation:

4m - (30cm + 40mm)

Converting cm and mm to metre by dividing by 100 and 1000 respectively

=> 4.000m - (30/100 m + 40/1000 m)

=> 4.000m - (0.030m + 0.004m)

=> 4.000m - 0.034m

=> 3.966m

Answer:

3.66m

Step-by-step explanation:

First, we have units measured in meters, centimeters, and millimeters.  This means we have to convert everything to the same measurement.

The easiest way is to convert everything to meters, as that's what the unit in the final answer will be.

To convert centimeters to meters, divide by 100

30/100=0.3

To convert millimeters to meters, divide by 1,000

40/1000=0.04

Next, plug the values back into the original equation:

4m-(0.3+0.04)

solve the parenthesis first

4-0.34

3.66

So, this equals 3.66 meters.

Hope this helps! :)

The total differential of the function z = 7x^4y^5 is dz is (28x^3y^5)dx + (35x^4y^4)dy.

To find the total differential of the function z = 7x^4y^5, we need to compute the partial derivatives with respect to x and y, and then express dz in terms of dx and dy.

Computing the partial derivative with respect to x,
∂z/∂x = 4 * 7x^3y^5 = 28x^3y^5

Computing the partial derivative with respect to y,
∂z/∂y = 5 * 7x^4y^4 = 35x^4y^4

Express dz in terms of dx and dy,
dz = (∂z/∂x)dx + (∂z/∂y)dy
dz = (28x^3y^5)dx + (35x^4y^4)dy

So, the total differential of the function z = 7x^4y^5 is dz = (28x^3y^5)dx + (35x^4y^4)dy.

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Answer:

x ~= 61.0°

Step-by-step explanation:

This is a trig question. The marked angle is unknown so we will use an inverse trig function to find a missing angle. On your calculator you may need to use the 2nd button or shift to get not just "cos" but the inverse cos. It might look like cos with a -1 exponent or "arccos" or "acos"

So the triangle is a right triangle. The angle is next to one of the given sides. "Next to" is called "adjacent" these mean the same thing. One side is adjacent to the angle. The other given side is the hypotenuse.

The trig function that puts together Adjacent and Hypotenuse is cosine.

cos (angle) = adj/hyp

We know the two sides. Fill them in.

cos (angle) = 31/64

We are looking for the angle.

cos x = 31/64

Use the inverse cos to find the angle.

cos^-1 (31/64) = x

This is a calculator activity. Unless your teacher has given you tables of values (this is how it was done before calculators) you have to use a calculator.

x = 61.0284677763

round to the nearest tenth.

x ~= 61.0