A boat can be hired for children's parties. The
cost of hiring the boat can be worked out using
the formula below.
The cost of a party was £314.50. How many
children were at the party?

Cost (£) £12.50 × the number of children + £27
=

Limaçons, Cardioids, and other polar curves are curves described in polar form by their radial distance from the origin and the angle they make with a fixed reference line

In polar form, curves are described in terms of their radial distance from the origin and the angle they make with a fixed reference line.

Here are the definitions of some common polar curves:

Limaçons: A limaçon is a polar curve defined by the equation r = a + b cos(θ) or r = a + b sin(θ), where a and b are constants. The shape of the limaçon depends on the values of a and b. If a > b, the curve has a loop that encloses the origin; if a = b, the curve is a cardioid; and if a < b, the curve has a dimple that encloses the origin.

Cardioids: A cardioid is a special case of a limaçon where a = b. The equation of a cardioid is r = a + a cos(θ) or r = a + a sin(θ), where a is a constant. A cardioid looks like a heart-shaped curve.

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Answer:

The answer is x = 2/7

Step-by-step explanation:

To find the intersection between 6x - 2 and 2 - 8x, we need to solve the equation:

6x - 2 = 2 - 8x

Adding 8x to both sides, we get:

14x - 2 = 2

Adding 2 to both sides, we get:

14x = 4

Dividing both sides by 14, we get:

x = 4/14 = 2/7

To find the values of x that satisfy the inequality 7x + 6 ≤ 13, we need to solve the inequality:

7x + 6 ≤ 13

Subtracting 6 from both sides, we get:

7x ≤ 7

Dividing both sides by 7, we get:

x ≤ 1

Therefore, the intersection between 6x - 2 and 2 - 8x for values of x that satisfy 7x + 6 ≤ 13 is x = 2/7. Since 2/7 is less than or equal to 1, it satisfies the inequality.

So the answer is x = 2/7.

Hope this helps! Sorry if it's wrong. If you need more help, ask me! :]

Answer:

Step-by-step explanation:

x = 4 or x = -2

Answer:

[tex]x=\frac{b-a}{ab}[/tex]

Step-by-step explanation:

[tex]\frac{1}{ax} -\frac{1}{bx} =1\\\frac{1}{ax} *\frac{b}{b} -\frac{1}{bx} *\frac{a}{a} =1\\\frac{b}{abx} -\frac{a}{abx}=1\\\frac{b-a}{abx}=1\\b-a=abx\\x=\frac{b-a}{ab}[/tex]

Answer:

z = 14

3x + 2y = 30

Step-by-step explanation:

z = 14

3x + 2y = 30 are the equations because they have an equal sign, =, in them.

The others are "expressions" because they don't have the equal sign.

Answer:

74 + 4 π ft^2

Step-by-step explanation:

Just a bit of a piecemeal.

2 rectangles and a half circle. We have all the dimensions.

1. Let's solve the big one first A = lw, putting in the numbers A = 4*15 = 60

2. The smaller one A = lw, putting int eh numbers A = 2 * 7 = 14

3. Now, the semi-circle. We know the diameter by deducting 12 from the total length, which is 9 + 7 = 16 in = 4 ft

The radius of the circle is half of the diameter, so the radius is 2 ft

The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.

Substituting the values, we get:

A = π(2)^2

A = 4π

The area of the circle is 4π square ft^2

Add: 60 + 14 + 4π = 74 + 4 π ft^2

According to the distributive property of algebraic expressions, each term in an expression's sum or difference must be multiplied by a number outside of the parenthesis. A number is used as the value outside of the parenthesis, with the total or difference.

By employing the distributive property of multiplication in step 2, we can reduce the expression by multiplying 5 by both 3 and y. So, our total is [tex]9 + 15 + 5y[/tex] .

In step 3, we use the commutative characteristic of addition to reorder the terms in the phrase. We now have  [tex]15 + 9 + 5y,[/tex] which equals [tex]24 + 5y[/tex] .

Step 4 involves applying the commutative property of addition to rearrange the equation's terms once more. This leads to the final simplified formulation, which is [tex]5y + 24[/tex] .

Therefore, The justification offered in step 1 is provided expression because it is the first expression provided and doesn't need to be further explained.

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It will take approximately 7.78 years for your salary at job A to exceed your salary at job B.

To solve this problem, we need to set up an equation. Let x represent the number of years that it will take for your salary at job A to exceed your salary at job B. We can use this equation to solve for x:

$46,000(1.05)^x > $55,000(1.02)^x

Now, let's simplify this equation to solve for x:

(1.05)^x > (1.02)^x * (55000/46000)

Taking the natural log of both sides of the equation:

xlog(1.05) > log(1.02) + log(55000/46000)

Solving for x:

x > [log(1.02) + log(55000/46000)]/log(1.05) ~ 7.78

It will take approximately 7.78 years for your salary at job A to exceed your salary at job B.

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By answering the presented question, we may conclude that  For slope example, if a figure's measures are in metros, the volume will be in cubic metros (m3).

The slope of a line determines how steep it is. The gradient is expressed mathematically as gradient overflow. The slope is computed by dividing the vertical difference (run) between two locations by the height difference (rise). The curve form of an expression is used to describe the straight line equation, which is written as y = mx + b. The y-intercept of the line is located where the inclination is m, b is b, and (0, b). For example, consider the slope and y-intercept of the equation y = 3x - 7. (0, 7). The y-intercept is placed where the slope of the path is m, b is b, and (0, b).  

When we calculate the volume of a 3D figure, we call our solution "units 3 (cubed)".

The term "cubed" is used since volume is calculated by multiplying three dimensions (length, breadth, and height), resulting in a unit raised to the power of three. For example, if a figure's measures are in metres, the volume will be in cubic metros (m3).

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False. The area of a circle is equal to 7.854.

The formula for the circumference of a circle is C = 2πr,

where C is the circumference and r is the radius. We can rearrange this formula to solve for the radius:

r = C/2π.

In this case, we are given that the circumference is 10, so we can calculate the radius as:

r = 10/2π

r = 5/π

To calculate the area of a circle, we use the formula

[tex]A = \pi r^2[/tex]

Substituting the value we found for r, we get:

[tex]A = \pi (5/\pi )^2\\A = \pi (25/\pi^2)\\A = 25/\pi[/tex]

This is approximately equal to 7.9577, which is not equal to 100. Therefore, the statement "A circle with a circumference of 10 has an area of 100" is false.

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The duration of the compression event is  100 degrees of crankshaft rotation by using the number of degrees of crankshaft rotation between the point where the intake valve closes (IVC) and the point where the exhaust valve opens (EVO).

To determine the duration of the compression event, we need to find the number of degrees of crankshaft rotation between the point where the intake valve closes (IVC) and the point where the exhaust valve opens (EVO).

Intake valve opens (IVO) at 8 degrees before top dead center (BTDC)

Intake valve closes (IVC) at 50 degrees after bottom dead center (ABDC)

Exhaust valve opens (EVO) at 50 degrees before bottom dead center (BBDC)

Exhaust valve closes (EVC) at 8 degrees after top dead center (ATDC)

First, we need to determine the position of the piston at each of these valve events. We know that the stroke of the engine is 180 degrees, so we can use this information to calculate the position of the piston at each event:

IVO: piston is at 8 degrees BTDC

IVC: piston is at 180 - 50 = 130 degrees ATDC

EVO: piston is at 180 + 50 = 230 degrees ATDC

EVC: piston is at 360 - 8 = 352 degrees BTDC

To find the duration of the compression event, we need to calculate the number of degrees of crankshaft rotation between IVC and EVO. We can do this by subtracting the position of the piston at IVC from the position of the piston at EVO:

Duration of compression event = EVO - IVC

= 230 - 130

= 100 degrees

Therefore, the duration of the compression event is 100 degrees of crankshaft rotation.

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We need a sample size of 239 to estimate the population mean height for college students with 95% confidence interval and 1.2 cm margin of error.

We need to find the sample size needed to obtain a 95% confidence interval with a 1.2 centimeters margin of error.Given that population standard deviation of student height is 9.48 centimeters.Using the formula of margin of error, we getME = z* σ/√nHere, σ = 9.48, ME = 1.2, and confidence interval = 95%.Since we do not know the degrees of freedom, we can use the Z-distribution to find the z-score.For 95% confidence interval, α/2 = 0.025The z-score for the 0.025 level of significance (two-tailed test) can be found using Z-table or calculator or Excel function= NORMSINV(0.025) = -1.96ME = z* σ/√n1.2 = -1.96 * 9.48 /√n√n = 1.96 * 9.48 /1.2= 15.42n = (15.42)^2= 238.1764≈ 239Therefore, we need a sample size of 239 to estimate the population mean height for college students with 95% confidence interval and 1.2 cm margin of error.

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Answer:

orange box = [tex]1 \frac{1}{4}[/tex]

blue box = [tex]1\frac{3}{4}[/tex]

Step-by-step explanation:

We are counting up in quarters (1/4) so we add 1/4 (or a quarter) on every time.

1/4 + 1/4 = 2/4 = 1/2 (equivalent fractions)

2/4 + 1/4 = 3/4

3/4 + 1/4 = 4/4 = 1 whole = 1

1 + 1/4 = 1 1/4 = orange box

1 1/4 + 1/4 = 1 1/2 or 1 2/4

1 2/4 + 1/4 = 1 3/4 = blue box

etc

hope this makes sense.

The probability of getting a sum that is a divisor of 10 is:P(sum is a divisor of 10) = 4/36= 1/9.

The sample space for the experiment "Toss three coins" is given by:{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}b. The event, E: "Get exactly two heads" consists of the following elements:{HHT, HTH, THH}c. The probability of the event, E: "Get exactly two heads" is given by:P(E) = number of favorable outcomes / number of possible outcomes= 3/8For the experiment of rolling two dice, the sample space is given by the set of all possible ordered pairs {(i, j)}, where i and j are numbers from 1 to 6. Thus, the sample space contains 6*6 = 36 elements.a. For the sum to be 10, the pairs that satisfy the condition are (4, 6), (5, 5), and (6, 4). Thus, the probability of getting a sum of 10 is:P(sum = 10) = 3/36= 1/12b. For the sum to be a divisor of 10, we need the pairs (2, 8), (4, 6), (6, 4), and (8, 2). Thus, the probability of getting a sum that is a divisor of 10 is:P(sum is a divisor of 10) = 4/36= 1/9.

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The volume of the cone is 10.5 in³.

Given that,

The radius of the cone = 2.5 in

The height of the cone = 1.6 in

The volume of the cone = [tex]\frac{1}{3}\pi r^{2} h[/tex]

                                        = [tex]\frac{1}{3}[/tex] × 3.14 × (2.5)²× 1.6

                                        =[tex]\frac{1}{3}[/tex] × 3.14 × 6.25 × 1.6

                                        = 10.5 in³

Therefore, the volume of the cone = 10.5 in³.

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By answering the presented question, we may conclude that As a result, proportionality the loan debt after 90 days is roughly $1713.17.

Proportionate relationships are those that have the same ratio every time. For example, the average number of apples per tree defines how many trees are in an orchard and how many apples are in an apple harvest. Proportional refers to a linear relationship between two numbers or variables in mathematics. When the first quantity doubles, the second quantity doubles as well. When one of the variables decreases to 1/100th of its previous value, the other falls as well. When two quantities are proportional, it means that as one rises, the other rises as well, and the ratio between the two remains constant at all levels. The diameter and circumference of a circle serve as an example.

Let B represent the loan balance at any moment t. (t).

k * B d(B(t))/dt (t)

where k is a proportionality constant.

This differential equation may be solved by separating the variables.

k * dt = d(B(t))/B(t).

When both sides are combined, the following results:

B(t) ln(t) = k*t + C

where C is an integration constant.

ln(B(0)) = k*0 + C

ln(1600) = C

So,

[tex]k*t + ln(B(t)) = ln(B(t)) (1600)\\= k*1 + ln(B(1)) (1600)\\ln(1920) = k + ln (1600)\\k = ln(1920) - ln (1600)\\k = ln(1.2) (1.2)[/tex]

Therefore,

[tex]ln(B(t)) = ln(1600) * 1.2 * t\\1600 * 1.2t = ln(B(t))\\B(t) = 1600 * 1.2^t\\[/tex]

To calculate the loan balance after 90 days, enter t=90/365:

[tex]B(90/365) = 1600 * 1.2^(90/365)\\B(90/365) ≈ $1713.17\\[/tex]

As a result, the loan debt after 90 days is roughly $1713.17.

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The answer of the given question based on the rhombus ABCD finding PD the answer is  PD = -5.

In geometry, diagonal is  straight line segment that connects two non-adjacent vertices of polygon. A polygon is any two-dimensional shape with straight sides, like   triangle, rectangle, square, or any other n-sided figure.

In a rectangle,  diagonal is  line segment that connects two opposite corners of rectangle.

Let's label the points as shown in the diagram:

   A

   / \

  /   \

 /     \

D-------B

   P

We know that DB = 2x - 4 and PB = 2x - 9. We need to find PD.

Since  diagonals of  rhombus bisect with each other, we have:

PD = PB - BD

Substituting the given values, we get:

PD = (2x - 9) - (2x - 4)

Simplifying, we get:

PD = 2x - 9 - 2x + 4

PD = -5

Therefore, PD = -5.

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Complete question is

In rhombus ABCD, if DB = 2x - 4 and PB = 2x - 9, find PD.

the diagram is also provided in the answer. you can refer there.

In response to the question, we may say that  Line A has a slope of -4, which is lower than that of line B.

The slope of a line determines how steep it is. A mathematical expression for the gradient is gradient overflow. The vertical change (run) between two spots is divided by the height change (rise) between the same two locations to get the slope. The equation of a straight line, written as y = mx + b, is represented by the curve form of an expression. The inclination is m, b is b, and the line's y-intercept is located at these points (0, b). As an example, consider the y-intercept and slope of the equation y = 3x - 7. (0, 7). The y-intercept is situated when the slope of the path is m, b is b, and (0, b).

With the two provided points, we must first establish whether of the following claims is true:

Slope of line B = (Y) / (X) = (-11 - 1) / (1 - (-2)) = -12 / 3 = -4

As a result, we can identify the following claims as being true:

The slope of line B is -5. (False)

The slope of line A is -4. (Really) The slope of lines A and B is the same. (False) Line B slopes more gently than line A. (False) Line A slopes more gently than line B. (True)

The following propositions are true:

Line A has a slope of -4, which is lower than that of line B.

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Part a: the success, with a probability of p = 1/5.

Part b:The mean of X is μ = np = 6(1/5) = 1.2.

Part c:The probability of at least two customers receiving the coupon can be computed using the binomial distribution formula, P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - [tex](6C1)(1/5)^1(4/5)^5 - (6C0)(1/5)^0(4/5)^6[/tex]

Part A: Yes, X is a binomial random variable. A binomial random variable is the number of successes in a sequence of n independent trials, where each trial has a probability p of success. In this case, X is the number of customers that receive the 50% off coupon, which is the success, with a probability of p = 1/5. There are also a total of n = 6 independent trials, which is the number of customers that made purchases from the website.

Part B: The mean of X is μ = np = 6(1/5) = 1.2. This means that, on average, the store can expect 1.2 customers to receive the 50% off coupon. The standard deviation of X is σ = √(np(1 - p)) = √(6(1/5)(1 - 1/5)) = 0.9. This means that there is a large degree of variability in the number of customers that receive the 50% off coupon.

Part C: The store's claim is accurate. The probability of at least two customers receiving the coupon can be computed using the binomial distribution formula, P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - [tex](6C1)(1/5)^1(4/5)^5 - (6C0)(1/5)^0(4/5)^6[/tex]

≈ 0.477,

which is close to the claimed probability of 1/5.

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Answer:

x≤4

Step-by-step explanation:

a) The function S(t) that inputs the number of minutes since the ice was removed from the freezer and outputs the side length of the cube in the salt is S(t) = 3 - 0.5t cm.

b) S(t) = 0 when t = 6 min, which means that the ice cube in the salt has melted completely.

c) The function P(t) that inputs the number of minutes since the ice was removed from the freezer and outputs the side length of the other cube is P(t) = 3 - 0.1t cm.

d) The function V(x) that inputs the side length x of an arbitrary cube and outputs the volume of that cube is V(x) = x^3 cm^3.

e)

(i) P(2) makes sense and represents the side length of the other cube 2 min after it was removed from the freezer.

(ii) V(S(5)) makes sense and represents the volume of the cube in the salt 5 min after it was removed from the freezer. (iii) P(S(1)) does not make sense because the P and S functions take a time value, not a side length.

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Answer: [tex]\frac{10}{3}[/tex]

Step-by-step explanation:

7+3(5-4)=10

2+1=3

Therefore, 10/3.

Given:-

[tex] \: [/tex]

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

put the given values in the equation

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

hope it helps ⸙

The solution to the expression (12x³ + 9x² - 3) / 3x is (4x³ + 3x² - 1) / x

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

Given the expression:

(12x³ + 9x² - 3) / 3x

To solve, we need to factorize the numerator to get:

= 3(4x³ + 3x² - 1) / 3x

= (4x³ + 3x² - 1) / x

The solution to the expression is (4x³ + 3x² - 1) / x

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The answer of the given question based on the right triangle the answer is, the right triangle has side measures of WX ≈ 16.8, XY ≈ 28.5, and XZ ≈ 32.5, and angle measures of m∠W = 33°.

A angle is geometric figure formed by two rays that share common endpoint called  vertex. The two rays are called  sides or legs of  angle, and angle is typically denoted by vertex letter, with small arc between  two sides to indicate angle.

Let's start by labeling the sides and angles:

   /|

  / |

 /  |

WX/___| Z

Y

WX is opposite to angle W

XY is adjacent to angle W

XZ is the hypotenuse of the triangle

Using the given information, we know that:

m∠W = 33°

XZ = WX / sin(W) (using the sine ratio)

We can solve for XZ as follows:

XZ = WX / sin(W)

XZ = XY / cos(W) (using the complementary angle of 90° - 33° = 57°)

We don't know the length of XY, but we can find it using the Pythagorean theorem:

XY² + WX² = XZ²

XY² + WX² = (WX / sin(W))²

XY² = (WX / sin(W))² - WX²

XY = sqrt((WX / sin(W))² - WX²)

Plugging in the given values, we get:

XY = sqrt((WX / sin(33°))² - WX²)

With WX rounded to the nearest tenth, we get:

WX = 16.8

XY = 28.5

XZ = 32.5

Therefore, the right triangle has side measures of WX ≈ 16.8, XY ≈ 28.5, and XZ ≈ 32.5, and angle measures of m∠W = 33°.

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Answer:

Step-by-step explanation:

tan^2 x +1=sec^2 x

1= sec^2 x-tan^2 x

The equivalent rate measured in qt/wk is 5.25 qt/wk, which is closest to option C. 12 qt/wk.

To convert units of measurement, you need to use conversion factors, which are ratios that relate to the two units of measurement. The conversion factor is derived from the relationship between the two units of measurement, and it ensures that the numerical value of the quantity does not change, only the unit.

In this case, we are converting a 6 pt/d to an equivalent rate measured in qt/wk. We need to use conversion factors for both volume and time to do this.

First, we need to convert pints to quarts. Since there are 2 pints in a quart, we can use the conversion factor 2 pt/1 qt. Multiplying by this conversion factor gives:

Therefore, the equivalent rate measured in qt/wk is 5.25 qt/wk, which is closest to option C. 12 qt/wk.

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In a rectangle having an area of 36 cm, if the length is three times more than the width of the rectangle is approximately 3.46 cm.

Let's assume that the width of the rectangle is "w" cm-

According to the problem, the length of the rectangle is three times longer than the width. Therefore, the length of the rectangle would be 3w cm.

The area of the rectangle is given as 36 cm². We know that the formula for the area of a rectangle is A = length x width.

So, we can substitute the values we have and get:-

36 = (3w) x w

Simplifying the equation, we get:-

36 = 3w²

Dividing both sides by 3, we get--

12 = w²

Taking the square root of both sides, we get--

w = √12

w ≈ 3.46

Therefore, the width of the rectangle is approximately 3.46 cm.

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The manufacturer can maximize their revenue by producing 60 road bikes and 120 touring bikes. The maximum revenue of the manufacturer is $24,000.

Revenue is the total amount of money earned by a business from its activities. It is calculated by subtracting the cost of goods sold from the total sales of goods and services. Revenue indicates how well a company is doing financially and serves as a key indicator of a company's performance. Revenue is an important factor in calculating the company's profitability.

This satisfies the minimum requirements of the supply contract, while also utilizing all of the available resources, frames and labor.

The total cost of producing 60 road bikes and 120 touring bikes is 400 frames and 900 hours of labor. This leaves no frames or labor unused. The total revenue from selling these bicycles is $24,000 (60 x $400 + 120 x $200). Therefore, the maximum revenue of the manufacturer is $24,000.

To maximize their revenue, the manufacturer should produce 60 road bikes and 120 touring bikes. This will ensure that all of their resources are used and they will get the highest return on their investment. This will also help them meet the minimum requirements of the supply contract.

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The statement "the expression (213 + 55) ÷ 4 show how you could calculate the amount of money Ross has" is true. Dividing the total cost by 4 is the same as multiplying by 4. The correct answer is (c).

The expression (213 + 55) ÷ 4 represents the calculation of the total cost of the computer and software divided by four, which is the amount of money Ross has. The total cost of the computer and software is $213 + $55 = $268. Dividing $268 by 4 gives $67, which is a fourth of the total cost. Therefore, Ross has $67.

Dividing by 4 is the same as multiplying by 1/4. So, another way to write the expression is (213 + 55) × (1/4). Both expressions represent the same calculation and give the same result.

Therefore, option (c) is the correct answer, and the expression (213 + 55) ÷ 4 shows how to calculate the amount of money Ross has.

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Complete question is:

Ross has a fourth of the amount needed to buy a new computer. The computer costs $213 and the additional software costs $55. Does the expression (213 + 55) ÷ 4 show how you could calculate the amount of money Ross has? Explain.

a) Yes. Dividing the total cost by 14 is the same as multiplying by 14.

b) No. There is no way to tell how much money Ross has from this expression.

c) Yes. Dividing the total cost by 4 is the same as multiplying by 4.

d) Yes. Dividing the total cost by 4 is the same as multiplying by 14