The quantity of a substance can be modeled by the function R(t) that satisfies the differential equation dR/dt= -1/5(R – 20). One point on this function is R(2) = 35. Based on this model, use a linear approximation to the graph of Rat t = 2 to estimate the quantity of the substance at t 1.9. \

The median is a better center for this data because the data is not normally distributed and clusters on the left.

Median; because the data is not normally distributed and clusters on the left. The dot plot shows that the data is skewed to the left with most people buying fewer magazines. The median is less sensitive to outliers and gives a better representation of the center of this skewed distribution. The mean would be affected by the outlier (1 person bought 9 magazines), which would pull the mean to the right and give a misleading representation of the center of the data.

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The volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.

To find the volume of a regular pentagonal prism with an edge length of 9 m and a height of 13 m, follow these steps:

Step 1: Find the apothem (a) of the base pentagon. Use the formula a = s / (2 * tan(180/n)), where s is the edge length and n is the number of sides (5 for a pentagon).

a = 9 / (2 * tan(180/5))
a ≈ 6.1803 m

Step 2: Calculate the area (A) of the base pentagon. Use the formula A = (1/2) * n * s * a.

A = (1/2) * 5 * 9 * 6.1803
A ≈ 139.3541 m²

Step 3: Determine the volume (V) of the pentagonal prism. Use the formula V = A * h, where h is the height.

V = 139.3541 * 13
V ≈ 1811.6033 m³

So, the volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.

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Answer:

h = 24 cm

Step-by-step explanation:

Given:

C (base) = 60π cm

V (volume) = 21,600π cm^3

Find: h (height) - ?

[tex]c = 2\pi \times r[/tex]

[tex]2\pi \times r = 60\pi[/tex]

[tex]2r = 60[/tex]

[tex]r = 30[/tex]

We found the length of the radius

v = 1/3 × πr^2 × h

1/3 × π × 900 × h = 21600π

Multiply both sides by 3:

2700π × h = 64800π / : 2700π

h = 24 cm

a. The teacher's age in seconds can be written in scientific notation as 1.5 × [tex]10^{9}[/tex] seconds.

b. A more reasonable unit of measurement for this situation could be years, as it is a common unit used to express human age.

c. To convert the teacher's age from seconds to years, we can divide the number of seconds by the number of seconds in a year. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and approximately 365 days in a year. So,

1.5 × [tex]10^{9}[/tex] seconds ÷ (60 seconds/minute × 60 minutes/hour × 24 hours/day × 365 days/year) = approximately 47.5 years

Therefore, the teacher is approximately 47.5 years old when using the more reasonable unit of measurement.

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Ginny's family traveled 3/10 of the total distance to her aunt's house on Sunday.

Let's say the total distance to Ginny's aunt's house is represented by the fraction 1.

On Saturday, Ginny's family traveled 3/10 of the distance, leaving 7/10 of the distance remaining.

Then, on Sunday, they traveled 3/7 of the remaining distance.

To find out what fraction of the total distance was traveled on Sunday, we need to multiply the distance traveled on Saturday and Sunday and divide by the total distance:

So, we have

Fraction = 3/7 * 7/10

Evaluate

Fraction = 3/10

Therefore, Ginny's family traveled 3/10 of the total distance to her aunt's house on Sunday.

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Based on the given information, we need to determine whether the Mean Value Theorem can be applied to the dosed interval [100-V2, 14:21].

The Mean Value Theorem states that for a function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) where the slope of the tangent line to the function at c is equal to the average rate of change of the function over the interval [a, b].

In this case, we do not have enough information about the function or its continuity on the interval [100-V2, 14:21]. Therefore, we cannot determine whether the Mean Value Theorem can be applied or not.

However, we do know that for the Mean Value Theorem to be applicable, the function must be continuous on the closed interval. If the function is not continuous on the closed interval, then the Mean Value Theorem cannot be applied.

Therefore, the answer to the question is B. No, because we do not have enough information about the function's continuity on the dosed interval [100-V2, 14:21].

I understand you're asking about the Mean Value Theorem and whether it can be applied to a given interval. Due to some typos in your question, I'm unable to identify the specific interval and function. However, I can provide general guidance.

The Mean Value Theorem can be applied to a function if:
A. The function is continuous on the closed interval [a, b]
B. The function is differentiable on the open interval (a, b)

If the given function meets these two conditions, then the Mean Value Theorem can be applied. Otherwise, it cannot be applied.

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Hence, 25% is the likelihood that the mouse will succeed in escaping the maze if all turns are made at random.

The examination of random chance and the likelihood that they will occur is the focus of the mathematical field of probability. It is a gauge of how likely an event is to occur and is represented by a value between zero and 1. A probability of 1 indicates that an event will undoubtedly occur. The probability of an occurrence is zero if it cannot occur. An event's probability is 0.5, or 50%, when it possesses a 50/50 chance of occurring. The number of favourable outcomes is divided by the entire amount of possible results to determine probability.

given

Based on the tree diagram, the following is an orderly list of every route that might be taken:

Three left turns and one right turn, in whatever order, make up each route.

As there are two options (left or right) for each turn, there are a total of 24 = 16 potential sequences of four turns.

Only if the mouse makes precisely three left turns and one right out of these will it be able to escape the maze.

Three lefts and one right can be arranged in one of four distinct ways (LLL, LLR, LRL, RLL), so the likelihood that the mouse will elude the maze is:

P(escape) = Number of favourable results / Number of potential results = 4 / 16 = 0.25 = 25%

Hence, 25% is the likelihood that the mouse will succeed in escaping the maze if all turns are made at random.

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The limit of [tex]L_n[/tex] as n approaches infinity is 1/2, and it can be expressed as the definite integral of x from 0 to 1.

To express the limit of [tex]L_n[/tex] as n approaches infinity as a definite integral, we can use the fact that the limit of a Riemann sum is equal to the corresponding definite integral. Thus, we can rewrite [tex]L_n[/tex] as:

[tex]L_n[/tex] = 1/n * (0 + 1 + 2 + ... + (n-1))

This is a Riemann sum for the integral:

[tex]\int\limits^1_0 {x} \, dx[/tex]

with n subintervals of width 1/n. Therefore, we can write:

[tex]\lim_{n \to \infty} L_n = \lim_{n \to \infty} 1/n * (0 + 1 + 2 + ... + (n-1)) = \int\limits^1_0 {x} \, dx = [x^2/2] \ from \ 0 \ to \ 1 = 1/2[/tex]

So, the limit of Ln as n approaches infinity is 1/2, and it can be expressed as the definite integral of x from 0 to 1.

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Answer:

it should be 1/10 if that helps you

Answer:

0.01 does not have the same value as the others.

Step-by-step explanation:

10/100, 10%, and 1/10 is all the same. 0.01 is NOT the same because it is equal to 1/100, or 1% while the others were all equal to 10%. To figure this out, you must multiply 0.01 by 100 (or move the decimal point one space to the right two times) and we will get 1. This means that 0.01 is 1% which is also equal to 1/100, therefore it does not obtain the same value as the others.

The length of the diagonal is 12. 7in

To determine the length of the diagonal, we need to know the Pythagorean theorem.

The Pythagorean theorem states that the square of the hypotenuse side is equal to the sum of the squares of the other two sides of a triangle.

The other two sides are the opposite and the adjacent sides.

From the information given in the diagram, we have that;

The opposite side = 12in

The adjacent side = 4in

Substitute the values

x² = 12² + 4²

find the squares

x² = 160

find the square root

x = 12. 7 in

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According to the model, each additional year of education adds $1200 to a person's yearly income in their first job after completing schooling.

We're given the equation y = 1200x + 7000, where x represents the number of years of education, and y represents the yearly income.

To find how much 1 year of education adds to a person's yearly income, we need to look at the coefficient of the variable x, which is 1200.

So according to the model, 1 year of education adds $1,200 to a person's yearly income.

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we first need to find out how many street name boards can be purchased with a budget of R80,000. We can do this by dividing the budget by the cost of each street name board:

R80,000 ÷ R134 = 597.01

Since we cannot purchase a fraction of a street name board, we round down to the nearest whole number. Therefore, the municipality can purchase 597 new street name boards with a budget of R80,000.

To find out how much money will be left in the budget, we can subtract the cost of the street name boards from the initial budget:

R80,000 - (597 x R134) = R1,598

Therefore, the municipality will have R1,598 left in their budget after purchasing 597 new street name boards.

Street name boards are an important part of any municipality as they help residents navigate and locate specific areas. By budgeting for and installing new street name boards, the municipality is taking a proactive approach to improve their community's infrastructure and ensure the safety and well-being of their residents.

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To solve this differential equation using separation of variables, we can rearrange it as:

dx/(x^2 + 36) = (t^2 + 4)/dt

Now we can integrate both sides:

∫ dx/(x^2 + 36) = ∫ (t^2 + 4)/dt

To integrate the left side, we can use the substitution u = x/6, du/dx = 1/6 dx, and dx = 6 du:

∫ dx/(x^2 + 36) = ∫ du/u^2 + 1

= arctan(x/6) + C1

To integrate the right side, we can use the power rule:

∫ (t^2 + 4)/dt = (1/3)t^3 + 4t + C2

Putting these together, we have:

arctan(x/6) = (1/3)t^3 + 4t + C

Where C = C2 - C1 is the constant of integration.

Now we can solve for x:

x/6 = 6 tan((1/3)t^3 + 4t + C)

x = 36 tan((1/3)t^3 + 4t + C)

Using the initial condition 2(0) = 6, we have:

x(0) = 36 tan(C) = 6

tan(C) = 1/6

C = arctan(1/6)

Therefore, the solution to the differential equation with the given initial condition is:

x = 36 tan((1/3)t^3 + 4t + arctan(1/6))

First, let's rewrite the equation using the given terms and separating the variables:

(t^2 + 4) dx/dt = (x^2 + 36)

Now, separate the variables:

dx/x^2 + 36 = dt/t^2 + 4

Next, we'll integrate both sides:

∫(1/(x^2 + 36)) dx = ∫(1/(t^2 + 4)) dt

Using the substitution method, we find:

(1/6) arctan(x/6) = (1/2) arctan(t/2) + C

Now, we'll use the initial condition 2(0) = 6 to find the value of C. Since 2(0) = 0, we have:

(1/6) arctan(6/6) = (1/2) arctan(0/2) + C

This simplifies to:

(1/6) arctan(1) = C

Therefore, the solution to the differential equation is:

(1/6) arctan(x/6) = (1/2) arctan(t/2) + (1/6) arctan(1)

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Cathy served 32 more people on Tuesday than on Monday.

Given, on Monday, Cathy served 9 tables with 6 people at each table. On Tuesday, Cathy served 86 people. We have to find the number of people she served more on Tuesday than on Monday.

So, on Monday she served = 9 tables x 6 people per table

= 54 people.

To find out how many more people Cathy served on Tuesday than on Monday, we can subtract the number of people served on Monday from the number served on Tuesday.

i.e. 86 - 54 = 32.

Therefore, Cathy served 32 more people on Tuesday than on Monday.

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0.180 is the sample mean weight of grapes, and what is the margin of error

The sample mean weight is the midpoint of the confidence interval:

sample mean = (lower limit + upper limit) / 2 = (15.875 + 16.595) / 2 = 16.235

Therefore, the sample mean weight of grapes is 16.235 ounces.

The margin of error is half of the width of the confidence interval:

margin of error = (upper limit - sample mean) = (16.595 - 16.235) / 2 = 0.180

Therefore, the margin of error is 0.180 ounces.

So the correct answer is: "The sample mean weight is 16.235 ounces, and the margin of error is 0.180 ounces."

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Given that 27.5 is the bottom end of the box, the lower quartile's value equation (Q1) must fall between 27.5 and 30. Hence, the appropriate response is 37.

Since, A mathematical equation links two statements and utilizes the equals sign (=) to indicate equality.

In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions.

For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

Here, 25% of the data fall inside the lower quartile (Q1), which is represented by that number. Q1 is situated near the bottom of the box in the box plot.

According to the description, the box's boundary is at 30, and its size ranges from 27.5 to 42.5 on the number line. As a result, the median value of the middle 50% of the data is 37, with a range of 27.5 to 42.5.

There are some data points outside the middle 50% since the lines outside the box finish at 15 and 55.

Hence; 27.5 is the bottom end of the box, the lower quartile's value (Q1) must fall between 27.5 and 42.5.

Hence, the appropriate response is,

= 37

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If f'(x) = 8x³ + 12x + 2 and f(1) = -4, f(-1) is equal to -18.

Given that f'(x) = 8x³ + 12x + 2, we can find the original function f(x) by integrating f'(x) with respect to x:

f(x) = 2x⁴ + 6x² + 2x + C, where C is an arbitrary constant.

We can then use the given initial condition f(1) = -4 to solve for C:

f(1) = 2(1)⁴ + 6(1)² + 2(1) + C = -4

Simplifying, we get:

C = -16

Therefore, the function f(x) is:

f(x) = 2x⁴ + 6x² + 2x - 16

To find f(-1), we substitute x = -1 into the expression for f(x):

f(-1) = 2(-1)⁴ + 6(-1)² + 2(-1) - 16 = -18

Thus, f(-1) equals -18.

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Answer:

2/5

Step-by-step explanation:

To convert a percentage to a fraction, we divide the percentage by 100 and simplify the resulting fraction. For example, to convert 40% to a fraction, we divide 40 by 100 to get 0.4 and then simplify the fraction 2/5. Therefore, 40% is equal to 2/5.

The house's real measurements are 18 feet by 20 feet.

In everyday speech, a dimension is a measurement of an object's length, width, and height, such as a box.

The idea of dimension in mathematics is an expansion of the concepts of one-dimensional lines, two-dimensional planes, and three-dimensional space.

Examples of dimensions include width, depth, and height.

One dimension is that of a line, two dimensions are those of a square, and three dimensions are those of a cube. (3D).

So, scaling is the process of changing a figure's size to produce a picture.

Considering that a scale of 6 cm equals 12 ft.

Hence:

9 cm = 9 cm * (12 ft. per 6 cm) = 18 feet

10 cm = 10 cm * (12 ft. per 6 cm) = 20 feet

Therefore, the house's real measurements are 18 feet by 20 feet.

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Correct question:

A scale drawing of a house shows 9cm x10cm. If 6cm=12 ft, what are the actual dimensions?

Answer: 36

Step-by-step explanation:

[tex]\frac{1}{3} \pi 15^{2} h=2700\pi \\75h=2700\\h=36[/tex]

The singer sold approximately 512 music downloads for  17p and 456 CDs for 35p and earned a total profit of £246. 64.

Given data:

Total profit = £246. 64

The price for each music download is = 17p

Number of CDs sold = 456 CD

Price for each CD = 35p

Assume that the singer sold x music downloads. when she earns 17p for each music download then her total earnings from music downloads will be 0.17x. The total earnings from selling the CD are,

= 0.35 × 456

= 159.6.

From the above data, we can write the equation to find the value of x by,

0.17x + 159.6 = 246.64

0.17x = 246.64 - 159.6

0.17x = 87.04

x = 87.04/0.17

x = 512

Therefore, the singer sold approximately 512 music downloads.

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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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Answer:

[tex]\sf \bf \dfrac{-1}{4} \ ; \ \dfrac{-1}{2} \ ; \ \dfrac{-3}{4}[/tex]

Step-by-step explanation:

Arithmetic sequence:

     Each term in the arithmetic sequence is obtained by adding or subtracting a common number with the previous term.

To find the next three terms, we need to find the common difference.

         Common difference = second term - first term

                                            [tex]\sf = \dfrac{1}{2}-\dfrac{3}{4}\\\\=\dfrac{2-3}{4}\\\\=\dfrac{-1}{4}\\\\\\\text{Each term is obtained by adding $\dfrac{-1}{4} $ with the previous term}[/tex]

Next three terms are,

           [tex]\sf 0 + \left(\dfrac{-1}{4}\right)= 0 - \dfrac{1}{4}=\dfrac{-1}{4}\\\\\\\dfrac{-1}{4}+\left(\dfrac{-1}{4}\right)=\dfrac{-1}{4}-\dfrac{1}{4}=\dfrac{-2}{4}=\dfrac{-1}{2}\\\\\\\dfrac{-1}{2}+\left(\dfrac{-1}{4}\right)=\dfrac{-1}{2}-\dfrac{-1}{4}=\dfrac{-2-1}{4}=\dfrac{-3}{4}[/tex]

Answer:

π(70)(√(70^2 + 50^2)) = π(700√74) m^3

= 18,918 m^3

650 adult tickets and 350 child tickets were sold for the local high school musical.

To determine how many adult and child tickets were sold for the local high school musical, you can use a system of linear equations. Let's use the terms "x" for adult tickets and "y" for child tickets.

1. The total number of tickets sold is 1000, so we have:
x + y = 1000

2. The total amount collected from ticket sales is $7100, so we have:
8.50x + 4.50y = 7100

Now, let's solve this system of equations step-by-step:

Step 1: Solve the first equation for x:
x = 1000 - y

Step 2: Substitute the expression for x from Step 1 into the second equation:
8.50(1000 - y) + 4.50y = 7100

Step 3: Simplify the equation:
8500 - 8.50y + 4.50y = 7100

Step 4: Combine like terms:
-4.00y = -1400

Step 5: Divide both sides by -4.00:
y = 350

Step 6: Substitute the value of y back into the expression for x from Step 1:
x = 1000 - 350

Step 7: Calculate the value of x:
x = 650

So, 650 adult tickets and 350 child tickets were sold for the local high school musical.

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The answer is 96 inches, which is equivalent to 8 feet.

To find the length of the top edge of the yield sign, we need to use the formula for the area of a trapezoid:

A = (b1 + b2)h/2

where A is the area of the trapezoid, h is the height, b1, and b2 are the lengths of the two parallel bases of the trapezoid.

In this case, we are given the area of the sign (216 square inches) and the height (18 inches), but we don't know the length of either base. However, we do know that the shape of a yield sign is that of a regular octagon, which means it has eight equal sides and eight equal angles.

If we draw a line from the top of the sign to the midpoint of one of the sides, we will form a right triangle with the height of the sign as one leg, half the length of the top edge as the other leg, and the length of one of the sides as the hypotenuse. We can use the Pythagorean theorem to find the length of the side:

a^2 + b^2 = c^2

where a is the height of the sign (18 inches), b is half the length of the top edge (what we are trying to find), and c is the length of one of the sides.

Since the sign has eight sides, we can divide the total area by 8 to get the area of one of the eight triangles that make up the sign. We can then use this area to find the length of one of the sides:

A = (bh)/2

216 sq. in. = (bh)/2

432 sq. in. = bh

Since the sign is a regular octagon, each of the eight triangles has the same base (the side of the octagon) and height (half the length of the top edge), so we can use this equation to solve for b:

432 sq. in. = b(18 in.)/2

b = 48 in.

Now we know that half the length of the top edge is 48 inches, so the full length of the top edge is:

2(48 in.) = 96 in.

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Answer:

Step-by-step explanation:

eeee

To find the probability of Sarah cleaning her room or doing her homework, we can use the addition rule for probabilities. However, we need to be careful not to count the probability of Sarah cleaning her room and doing her homework twice. Therefore, we need to subtract the probability of Sarah cleaning her room and doing her homework from the sum of the probabilities of Sarah cleaning her room and doing her homework separately.

Let C be the event that Sarah cleans her room, and let H be the event that Sarah does her homework. Then we know:

P(C) = 0.40 (the probability that Sarah cleans her room)

P(H) = 0.70 (the probability that Sarah does her homework)

P(C and H) = 0.24 (the probability that Sarah cleans her room and does her homework)

Using the addition rule, we can find the probability of Sarah cleaning her room or doing her homework as follows:

P(C or H) = P(C) + P(H) - P(C and H)

          = 0.40 + 0.70 - 0.24

          = 0.86

Therefore, the probability of Sarah cleaning her room or doing her homework is 0.86, or 86%.

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To find the coordinates of point B, we can use the section formula which states that the coordinates of the point that divides a segment with endpoints (x1, y1) and (x2, y2) in the ratio of m:n are given by:

((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

The coordinates of point B are (4.25, 8.25), and the answer is (D).

Here, A (2, 6) and C (5, 9) are the endpoints of the segment, and we want to partition the segment in the ratio of 3:1. So, we have:

m:n = 3:1

m+n = 4

Solving for m and n, we get:

m = 3, n = 1

Now, substituting  values in the section formula, we get:

((35 + 12)/(3+1), (39 + 16)/(3+1)) = (4.25, 8.25)

Therefore, the coordinates of point B are (4.25, 8.25), and the answer is (D).

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Answer:

(4.25, 8.25)

Step-by-step explanation:

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