Find dy/dt given that x^2+y^2 = 2x+4y, x = 3, y = 1 and dx/dt = 7

The average price of milk in 2018 was 6.45 dollars per gallon.

The average price of milk in 2021 was 189.95 dollars per gallon.

The given function is: Price = 3.55 + 2.90(1 + x)³

In order to find the average price of milk in 2018, we need to set x = 0:

Price in 2018 = 3.55 + 2.90(1 + 0)³ = 3.55 + 2.90(1) = 6.45 dollars per gallon

Price in 2021 = 3.55 + 2.90(1 + 3)³ = 3.55 + 2.90(64) = 189.95 dollars per gallon

Hence, the average price of milk in 2021 was 189.95 dollars per gallon.

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Answer:

Pens=13

Pencils=27

Step-by-step explanation:

1. Pens = x, Pencils = y

x+y=40

2. y+5 & x-5 --> y=4x

3. Rearrange the x+y=40 to y=40-x. Then substitute y=4x and y=40-x to 4x=40-x.

4. Solve. 4x=40-x, 5x=40, x=8

5. Plug in x=8 to y=4x -> y=32.

6. Reverse the y+5 & x-5. y=27, x=13.

Answer:

1. 5,25,125,625,3125, 15625,78125

Rule: We multiply each term of the sequence by 5.

2.12, 24, 48, 96, 192, 384, 768

Rule: We multiply each term of the sequence by 2.

3. 85, 80, 75, 70, 65, 60, 55

Rule:subtract 5 from each term of the sequence

4.3;5,7,9,11,13,15

Rule: Add 2 to every term in the sequence

5.12, 13,15,16,18,19,21,22

Rule: We first add 1 to the previoys term then add 2 to the new term of the sequence and so on.

The fraction of Juan's shifts with a total sales of $225 or more can be found by looking at the box plot.

We can see that the top line of the box represents the third quartile (Q3) which is the value where 75% of the data falls below.

In this case, Q3 is at approximately $250. This means that 75% of Juan's shifts had total sales less than $250. To find the fraction of shifts with sales of $225 or more, we need to determine how many shifts fall within the range of $225 to $250.

Looking at the box plot, we can see that the distance between Q1 and Q3 (the interquartile range) is approximately $100. Therefore, the distance between Q1 and $225 is approximately one-third of the interquartile range or $33.33. So, any shift with total sales of $225 or more would fall within one-third of the distance between Q1 and Q3.

Therefore, the fraction of Juan's shifts with total sales of $225 or more is approximately one-third of 75%, which is 25%.

In summary, approximately 25% of Juan's shifts at the Bayview community pool had total sales of $225 or more, based on the box plot.

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Answer:

the answer for this problem would be -20

Answer:

-20

Step-by-step explanation:

The easiest way to do it is to change -3 3/7 and 5 5/6 into improper fractions. To do that, let’s take -3 3/7 for example. You would add the numerator by the whole number, and then multiply the denominator by the whole number. Getting you -24/7, the other number would be 35/6 then you multiply the two getting you -840/42, which it turned back into a proper fraction by dividing the two numbers, you would get -20. Hope this helps!

The only value of x where f'(x) = 0 is x = 0.

Let's find all values of x where the derivative of f(x) = [tex]arcsin(e^(2x) – 2x)[/tex] is equal to 0.

Step 1: Find the derivative f'(x) using the chain rule.

For this, we'll need to differentiate [tex]arcsin(u)[/tex] with respect to u, which is [tex](1/√(1-u^2))[/tex], and then multiply by the derivative of u [tex](e^(2x) – 2x)[/tex]with respect to x. So, f'(x) = [tex](1/√(1-(e^(2x) – 2x)^2)) * d(e^(2x) – 2x)/dx[/tex]

Step 2: Find the derivative of e^(2x) – 2x with respect to x. Using the chain rule and the derivative of [tex]e^u: d(e^(2x) – 2x)/dx = 2e^(2x) – 2[/tex]

Step 3: Combine the derivatives. f'(x) =[tex](1/√(1-(e^(2x) – 2x)^2)) * (2e^(2x) – 2)[/tex]

Step 4: Set f'(x) equal to 0 and solve for x. [tex](1/√(1-(e^(2x) – 2x)^2)) * (2e^(2x) – 2) = 0[/tex]

Since the first part of the product [tex](1/√(1-(e^(2x) – 2x)^2))[/tex] is never 0, we can focus on the second part: [tex]2e^(2x) – 2 = 0[/tex]

Step 5: Solve for x. [tex]2e^(2x) = 2 e^(2x) = 1[/tex]

The only way this is true is when 2x = 0, since [tex]e^0 = 1: 2x = 0 x = 0[/tex]

So, the only value of x where f'(x) = 0 is x = 0.

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The p-value of the test is 0.017 > 0.01, which means that there is not convincing evidence that the mean time behind the whell for all U.S drivers is less than 51 minutes.

We are told that:

Time spent by U.S. drivers on average each day = 51 minutes behind the wheel

Thus, the null hypothesis is:

H₀: μ = 51

The alternate hypothesis is:

Hₐ: μ < 51

The formula for the test statistic is:

z = (x' - μ)/(σ/√n)

where:

x' is sample mean

n is sample size

μ is population mean

σ is standard deviation

Thus:

z = (46.4 - 51)/(18.8/√75)

z = -2.12

From online p-value from z-score calculator, we have:

The p-value of the test is referred to as the probability of finding a sample mean that is less than 46.4, which is the p-value of z when X = 46.4.

Looking at the z-table, z = -2.12 has a p-value of 0.017.

Decision:

The p-value of the hypotheses test is seen to be0.017 > 0.01.

Thus, this tells us that we don't have convincing evidence that the mean time spent behind the wheel for all U.S drivers is less than 51 minutes.

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Megha's total movement involves biking 20km north, 30km east, 20km south, and 30km west, resulting in a displacement of zero as she ends up back at her starting point.

Given that,

Megha bikes 20km north.

Megha then bikes 30km east.

After that, Megha bikes 20km south.

Lastly, Megha bikes 30km west.

Megha stops after completing the above movements.

Megha's displacement can be calculated by finding the straight-line distance between her starting point and ending point.

In this case,

She initially bikes 20km north, then 30km east, followed by 20km south, and finally 30km west.

Let's break it down:

The north and south distances cancel each other out, as she ends up back at her starting point vertically.

The east and west distances also cancel each other out, as she ends up back at her starting point horizontally.

Hence,

Megha's displacement is zero. She has returned to her original position.

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Using quadratic formula, the value of x in the quadratic equation are 1.47 and -0.85

To find the value of x, we can either use quadratic formula or factorization method.

8x² - 2 - 5x = 8

Let's rewrite the equation properly

8x² - 5x - 2 - 8 = 0

8x² - 5x - 10 = 0

a = 8, b = -5, c = -10

Using quadratic formula;

-b ±[√b² - 4ac / 2a]

-(-5) ±[√(-5)² - 4(8)(-10) / 2(8)]

x = 5+ √345 / 16 or x = 5 - √345 / 16

x = 1.47 or x = -0.85

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Rectangles are relatable to factors because of their areas and perimeter equations

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

Explaining why rectangles are relatable to factors

Rectangles are relatable to factors because of the following reasons

Perimeter = 2 * (Length + width)

Area = Length * width

This means that in calculating the areas and the perimeters of a rectangles, we make use of arithmetic expressions

These arithmetic expressions, when expanded form terms and factors of  expression

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The location where the fourth child will stop running in the relay race is 4/5 of the way.

To determine the location where the fourth child will stop running in the relay race, we need to find out the total distance covered by the first four children.

Since each child runs 1/5 of a mile, we can find the total distance for the first four children by multiplying their individual distances.

Total distance = (Number of children) * (Distance run by each child)
Total distance = 4 * (1/5)

Now, let's multiply:
Total distance = 4/5

So, the fourth child will stop running at a location 4/5 of a mile from the starting point. To plot this point on a number line, locate the point between 0 and 1 that is 4/5 of the way. This point represents the location where the fourth child will stop running in the relay race.

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Percentage of gumballs that are within standard deviations of the mean is 68% and have a diameter between 2.20 cm and 2.22 cm.

To find the percentage of gumballs that are within one standard deviation of the mean, we need to use the empirical rule, also known as the 68-95-99.7 rule. According to this rule, for a normal distribution:

Here we want to find the percentage of gumballs that are within one standard deviation of the mean. So we can use the first part of the empirical rule and say that approximately 68% of the gumballs have a diameter between:

Mean - Standard deviation = 2.21 - 0.01 = 2.20 cm and Mean + Standard deviation = 2.21 + 0.01 = 2.22 cm

Therefore, approximately 68% of the gumballs have a diameter between 2.20 cm and 2.22 cm.

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The only option left is −5°F, which is colder than −3°F and thus could be the temperature on her thermometer this morning.

The temperature on Skylar's thermometer is measured using a scale that indicates the degree of hotness or coldness of the atmosphere. In this case, we know that the temperature on her thermometer last night was −3°F. The negative sign indicates that the temperature was below the freezing point of water.

We are asked to determine what the temperature could be on her thermometer this morning. Since we are told that the temperature is colder than −3°F, we can eliminate 0°F and 2°F as possible options, as they are warmer than −3°F.

Similarly, −1°F is only slightly colder than −3°F, so it is unlikely to be the temperature on Skylar's thermometer this morning. Therefore, the only option left is −5°F, which is colder than −3°F and thus could be the temperature on her thermometer this morning.

It is important to note that temperature can have a significant impact on our daily lives. Extreme cold temperatures can cause frostbite, hypothermia, and other health hazards, while hot temperatures can lead to dehydration, heat exhaustion, and heat stroke. It is important to dress appropriately and take necessary precautions when the temperature is too low or too high.

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To find dy/dx for y = 1/3x³ + 7x, we need to take the derivative with respect to x using the power rule and the sum rule, which gives dy/dx = x² + 7.

The given equation is y = 1/3x³ + 7x. To find dy/dx, we need to take the derivative of y with respect to x. We can do this by using the power rule and the sum rule of differentiation.

Using the power rule, the derivative of x³ is 3x². Using the sum rule, the derivative of 7x is 7. Therefore, the derivative of y with respect to x is:

dy/dx = d/dx (1/3x³ + 7x)

= d/dx (1/3x³) + d/dx (7x)

= (1/3) d/dx (x³) + 7 d/dx (x)

= (1/3) (3x²) + 7

= x² + 7

Hence, we have found that the derivative of y with respect to x is x² + 7.

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Ms. Paquette used approximately 0.73 gallons of paint for one coat on the rectangular wall.

To determine how many gallons of paint Ms. Paquette used for one coat on the rectangular wall, we first need to calculate the total area of the wall.

Area of the wall = length x width
= 25.5 feet x 10 feet
= 255 square feet

Now, to find out how many gallons of paint are needed for one coat, we need to know the coverage rate of the paint. This information is usually provided on the paint can label. Let's assume that the coverage rate for the paint Ms. Paquette used is 350 square feet per gallon.

To calculate how many gallons of paint are needed, we can divide the total area of the wall by the coverage rate of the paint:

255 square feet ÷ 350 square feet per gallon = 0.7286 gallons

This means that Ms. Paquette used approximately 0.73 gallons of paint for one coat on the rectangular wall. Since paint is typically sold in 1-gallon containers, she would need to purchase at least one gallon of paint to complete one coat on this wall.

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(1) it takes about 10 hours to thaw a 20-pound turkey in cold water.(2) it takes an additional 0.25 hours (or 15 minutes) to thaw in the refrigerator.

A linear function is a function that represents a straight line in the coordinate plane. For example, y = 3x - 2 represents a straight line in the coordinate plane and thus a linear function. Since y can be replaced by the function f(x), this function can be written as f(x) = 3x - 2.

 a. A typical thawing rate when thawing in the refrigerator  is about 1 day per 4-5 kilograms of turkey meat. So if you have a 20 pound turkey, it will take about 4-5 days to thaw in the fridge. In cold water, the thawing rate  is about 30 minutes per pound, so it takes about 10 hours to thaw a 20-pound turkey in cold water.

b) The initial value may represent the weight of the frozen turkey before thawing begins. c. Let y be the time it takes to thaw the turkey in hours and let x be the weight of the turkey in pounds. Assuming a linear relationship between melting time and weight, we can write:

y = mx + b

where m is the thaw rate (in hours per pound) and b is the intercept (the time it takes to thaw a 0-pound turkey, which is 0). From part a, we know that the refrigerator thaw rate  is about 1 day per 4-5 pounds of turkey, or about 0.25-0.2 hours per pound. So we can use m = 0.25 in our linear function:

y = 0.25x + 0

This means that for every additional pound of turkey, it takes an additional 0.25 hours (or 15 minutes) to thaw in the refrigerator.

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The relative frequency for selecting a teen out of all the people who like chocolate is calculated by dividing the number of teens who like chocolate (N) by the total number of people who like chocolate (T).

To find the relative frequency for selecting a teen out of all the people who like chocolate, you need to follow these steps:

Step 1: Determine the total number of people who like chocolate (let's call this T).
Step 2: Determine the number of teens who like chocolate (let's call this N).
Step 3: Calculate the relative frequency by dividing the number of teens who like chocolate (N) by the total number of people who like chocolate (T).

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The p-value is less than the significance level so reject the null hypothesis and concluded percentage of the students volunteer their time is different from receiving financial aid students.

Percentage of college students who volunteer their time = 20%

Perform a hypothesis test.

Null hypothesis H₀: p = 0.20,

where p is the proportion of college students who volunteer their time.

The alternative hypothesis is Hₐ: p ≠ 0.20.

Indicating that the proportion of college students who volunteer their time is different for students receiving financial aid.

57 out of 381 randomly selected students who receive financial aid volunteered their time.

Test the hypothesis,

Calculate the sample proportion of volunteers among the students receiving financial aid,

p₁ = 57 / 381

    = 0.149

Using Test statistic,

which follows a normal distribution under the null hypothesis .

Mean = 0  

Standard deviation σ = √(p(1-p)/n),

where p = 0.20 is the proportion under the null hypothesis

n = 381 is the sample size.

z

= (p₁ - p) /√(p×(1-p)/n)

= (0.149 - 0.20) / √(0.20(1-0.20)/381)

= -2.55

Using attached table of p-value from z-score.

Calculated test statistic of -2.55 corresponds to a p-value of 0.0054,

which is less than the significance level α = 0.01.

Reject the null hypothesis .

Therefore, we conclude that there is evidence to suggest that the percentage of college students who volunteer their time is different for students receiving financial aid.

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The above question is incomplete, the complete question is:

20% of all college students volunteer their time. is the percentage of college students who are volunteers different for students receiving financial aid? of the 381 randomly selected students who receive financial aid, 57 of them volunteered their time. what can be concluded at the α = 0.01 level of significance?

Answer: To calculate the total amount a buyer should save each month to cover these fees, we need to add up all the costs and divide by 12 (the number of months in a year).

Annual real estate taxes = $2046

Insurance premiums = $1236

Flood insurance premiums = $380

Annual special assessment landfill fee = $125

Quarterly trash collection costs = $65 x 4 = $260

Annual homeowner's association fee = $720

Quarterly pool maintenance costs = $45 x 4 = $180

Monthly termite and pest protection costs = $39 x 12 = $468

Total annual costs = $2046 + $1236 + $380 + $125 + $260 + $720 + $180 + $468 = $5405

Total monthly costs = $5405 / 12 = $450.42

Therefore, a buyer should save $450.42 each month to cover these fees.

Step-by-step explanation:

The shooter was located approximately 36.15 feet off the ground.

We can use trigonometry to solve for the height of the shooter.

First, we need to find the horizontal distance from the shooter to the man on the park bench. We can use the angle of 26° and the distance of 92 feet to calculate this distance:

horizontal distance = 92 feet * cos(26°)

horizontal distance = 82.33 feet

Next, we can use the height of 3.7 feet and the horizontal distance of 82.33 feet to find the height of the shooter:

tan(26°) = height difference / horizontal distance

height difference = horizontal distance * tan(26°)

height difference = 82.33 feet * tan(26°)

height difference = 36.15 feet

Therefore, the shooter was located approximately 36.15 feet off the ground.

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The critical points of f(x) are x = 0 and x = e^(-1/2) / 3, and x = e^(-1/2) / 3 corresponds to a local minimum of f(x). Cmin = e^(-1/2) / 3 and Cmax = 0.

Taking the derivative of f(x) with respect to x using the product rule and the chain rule, we get:

f'(x) = 14x ln(3x) + 7x

Setting f'(x) equal to zero and solving for x, we get:

14x ln(3x) + 7x = 0

Factor out x:

7x(2ln(3x) + 1) = 0

So either x = 0 or 2ln(3x) + 1 = 0.

If x = 0, then f'(x) = 0 and x is a critical point.

If 2ln(3x) + 1 = 0, then ln(3x) = -1/2 and 3x = e^(-1/2). Solving for x, we get:

x = e^(-1/2) / 3

So e^(-1/2) / 3 is also a critical point.

Now we need to apply the second derivative test to determine whether these critical points correspond to a local minimum or maximum.

Taking the second derivative of f(x), we get:

f''(x) = 14 ln(3x) + 21

For x = 0, we have:

f''(0) = 14 ln(0) + 21

The natural logarithm of zero is undefined, so the second derivative does not exist at x = 0. Therefore, we cannot apply the second derivative test at x = 0.

For x = e^(-1/2) / 3, we have:

f''(e^(-1/2) / 3) = 14 ln(1/e^(1/2)) + 21

= -14/2 + 21

= 7/2

Since the second derivative is positive at this point, we can conclude that x = e^(-1/2) / 3 is a local minimum of f(x).

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The best estimate of the volume of this cylindrical water dispenser is 2940 in³. The correct option is b.

To estimate the volume of the cylindrical water dispenser, we can use the formula for the volume of a cylinder, which is given by V = πr²h, where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius, and h is the height.

Given that the radius of the water dispenser is 7 inches and the height is 20 inches, we can substitute these values into the formula:

V = π(7²)(20)

V = π(49)(20)

V ≈ 3.14 × 49 × 20

V ≈ 3.14 × 980

V ≈ 3075.2 in³

From the given options, the closest estimate to the calculated volume of approximately 3075.2 in³ is option B: 2940 in³. While it is not an exact match, it is the closest estimate among the options provided.

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The circumference of the tank is is 125.66m and the percentage of the circumference that is visible to the person is 10.24%.

a) To calculate the circumference of the cylindrical tank, using the formula C = 2πr, where C is the circumference and r is the radius of the tank. In this case, r = 20 m, so:

C = 2π(20 m) ≈ 125.66 m

b) To determine the percentage of the circumference visible to the person, we first need to calculate the angle (in degrees) that subtends the visible arc. This can be done using the inverse tangent function:

angle = atan(opposite/adjacent) = atan(20 m/60 m)

angle ≈ 18.43°

Since the visible arc is symmetrical on both sides, the total angle of the visible arc is twice the calculated angle:

total angle ≈ 36.86°

Now, calculate the percentage of the circumference that is visible by dividing the total angle by 360° and multiplying by 100%:

percentage = (36.86°/360°) × 100% ≈ 10.24%

So, the visible percentage of the circumference to the person is approximately 10.24%.

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Answer:

Acute triangle

Step-by-step explanation:

angles that are less than 90° are called acute

This triangle has 3 acute angles, so it is an acute triangle

The formula for the area of a rectangle is length times width, or in this case (x+6)(x+2) or x^2 + 8x + 12 in standard form.

To simplify this expression, we can use the distributive property to multiply the two binomials:

(x+6)(x+2) = x(x+2) + 6(x+2)

= x² + 2x + 6x + 12

= x² + 8x + 12

To express the area of the entire rectangle, we need to multiply the length (x + 6) by the width (x + 2). This will give us a polynomial in standard form.

Step 1: Write down the expression for the area of the rectangle.
Area = (x + 6)(x + 2)

Step 2: Use the distributive property (also known as the FOIL method) to expand the expression.
Area = x(x + 2) + 6(x + 2)

Step 3: Continue to expand and simplify the expression.
Area = (x² + 2x) + (6x + 12)

Step 4: Combine like terms.
Area = x² + 8x + 12

So the area of the entire rectangle is expressed as the polynomial x² + 8x + 12 in standard form.

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The box plot that represents these data is Option B because the centre of measure falls on 4.

To find the median, we first arrange the study hours in ascending order:

2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7.

From here, we have 15 numbers, so the median is the 8th number in the list which is 4.

As the median study hours for Mr. Jeffer's math class is 4 hours per week, then, the box plot that represents these data is B because the centre of measure falls on 4.

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The group of natural numbers is a subset of the whole numbers.

What is Whole number ?

Whole numbers are a set of numbers that includes all positive integers (1, 2, 3, ...) and zero (0). They do not include negative numbers or fractions. Whole numbers are often used to count objects or represent quantities that cannot be divided into smaller parts.

Out of the given options, the group of natural numbers is the only one that is a subset of whole numbers. The other options - Rational numbers, Real numbers, Irrational numbers, and Integers - are not subsets of the whole numbers.

Rational numbers include fractions and decimal numbers, which can be expressed as a ratio of two integers, including non-whole numbers.

Real numbers include all rational and irrational numbers, including non-whole numbers. Irrational numbers are non-repeating, non-terminating decimals that cannot be expressed as a fraction. Integers include both positive and negative whole numbers, as well as zero.

The subset of whole numbers is called natural numbers.

Therefore, the group of natural numbers is a subset of the whole numbers.

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Answer:

I believe the distance between these points is (2, 7), because the difference between -5 and -7 is 2, and the for -9 and -2, it's 7.  Hope I helped.