volume of cylinders

The crate can hold 175 bricks.

First, we need to convert the volume of the crate from cubic meters to cubic centimeters because the volume of each brick is given in cubic centimeters.

1 m = 100 cm

Volume of crate = 2.8 m3 = 2.8 x (100 cm)3 = 2,800,000 cm3

Now we can find the number of bricks that can be packed into the crate by dividing the volume of the crate by the volume of each brick:

Number of bricks = Volume of crate / Volume of each brick

= 2,800,000 cm3 / 16,000 cm3

= 175 bricks

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Antonio and lizeth's adjusted gross income is $85,988.

To find their adjusted gross income, we need to start with their total income and subtract any adjustments.

Total income:

Combined income: $83,366

Interest income: $1,200

Rental income: $4,922

Total income before adjustments: $89,488

Adjustments:

Reduce income by $3,500

Adjusted gross income:

$89,488 - $3,500 = $85,988

Therefore, their adjusted gross income is $85,988.

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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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Answer:

Step-by-step explanation:

To find the experimental probability of a student in Homeroom 203 taking a certain number of minutes to get to school, we need to divide the number of students who take that amount of time by the total number of students in the homeroom.

The total number of students in the homeroom is:

3 + 5 + 10 + 7 + 2 + 3 = 30

The probability of a student taking less than 10 minutes to get to school is:

3/30 = 0.1 or 10%

The probability of a student taking 10-19 minutes to get to school is:

5/30 = 0.166 or 16.6%

The probability of a student taking 20-29 minutes to get to school is:

10/30 = 0.333 or 33.3%

The probability of a student taking 30-39 minutes to get to school is:

7/30 = 0.233 or 23.3%

The probability of a student taking 40-49 minutes to get to school is:

2/30 = 0.066 or 6.6%

The probability of a student taking 50 minutes or more to get to school is:

3/30 = 0.1 or 10%

To make a probability distribution, we can list the possible outcomes (in this case, the time it takes to get to school) and their corresponding probabilities:

Time (min) Probability

Less than 10 0.1

10-19 0.166

20-29 0.333

30-39 0.233

40-49 0.066

50 or more 0.1

Note that the probabilities add up to 1, which is what we expect for a probability distribution.

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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The parent functions of the function equations are x³, x⁴ and x²

The transformed functions 3 - 5 represent the given parameter

To derive the parent functions, we need to determine the degree of the transformed and use this degree as a guide

By definition, the degree of a function is the highest power in the function

So, we have

Question 3

g(x) = (1/2x + 2)³ + 5

The degree here is 3

This means that the function is a cube function

The parent function of a cube function is y = x³

So, the parent function is g(x) = x³

Question 4

g(x) = x⁴ - 4

The degree here is 4

This means that the function is a polynomial function shifted down by 4 units

The parent function of this is y = x⁴

So, the parent function is g(x) = x⁴

Question 5

g(x) = 1/2(x - 1)² - 4

The degree here is 2

This means that the function is a quadratic function

The parent function of a quadratic function is y = x²

So, the parent function is g(x) = x²

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Let x be the number of pinon nuts each animal has collected. To make the chipmunk have ten more pinon nuts than the squirrel, the squirrel would have to give the chipmunk 10 pinon nuts.

So, after the exchange, the squirrel would have x - 10 pinon nuts, and the chipmunk would have x + 10 pinon nuts.

Since they are each giving an equal amount, the total number of pinon nuts remains the same. Therefore, we can set up the equation:

x + (x - 10) = 2x - 10

Simplifying and solving for x, we get:

2x - 10 = 2x

-10 = 0

This is a contradiction, so there is no solution that satisfies the conditions of the problem.

Therefore, the problem is not well-defined and there is no answer.

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It will take 6 minutes for the water hose to fill the 2-gallon bucket at a rate of 1/3 gallon per minute.

To determine the time required to fill a 2-gallon bucket using a water hose that fills at a rate of 1/3 gallon per minute, you can use a simple calculation.

First, identify the fill rate of the hose, which is 1/3 gallon per minute. Now, consider the bucket's capacity, which is 2 gallons. To find out how many minutes it takes to fill the bucket, divide the total capacity of the bucket by the fill rate:

Time (minutes) = Bucket capacity (gallons) / Fill rate (gallons per minute)

In this case:

Time (minutes) = 2 gallons / (1/3 gallons per minute)

To solve this, you can multiply the numerator and denominator by the reciprocal of the fill rate:

Time (minutes) = 2 gallons * (3 minutes per gallon)

Time (minutes) = 6 minutes

So, it will take 6 minutes for the water hose to fill the 2-gallon bucket at a rate of 1/3 gallon per minute.

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The sphere must be emptied into the cylinder 3 times to completely fill it with water.

We will use the formulas for theSo, the sphere must be emptied into the cylinder 3 times to completely fill it with water. and the volume of a cylinder to find out how many times the sphere needs to be emptied into the cylinder until it is full.

Step 1: Find the volume of the sphere.
The formula for the volume of a sphere is V_sphere = (4/3)πr^3, where r is the radius.
Given that the radius of the sphere is 6 inches, we can calculate its volume:
V_sphere = (4/3)π(6)^3 = (4/3)π(216) ≈ 904.78 cubic inches

Step 2: Find the volume of the cylinder.
The formula for the volume of a cylinder is V_cylinder = πr^2h, where r is the radius and h is the height.
Given that the radius of the cylinder is 6 inches and the height is 18 inches, we can calculate its volume:
V_cylinder = π(6)^2(18) = π(36)(18) ≈ 2038.51 cubic inches

Step 3: Determine how many times the sphere must be emptied into the cylinder.
To find out how many times the sphere needs to be emptied into the cylinder, divide the volume of the cylinder by the volume of the sphere:

Number_of_times = V_cylinder / V_sphere = 2038.51 / 904.78 ≈ 2.25 times
Since we cannot empty the sphere partially, we'll round up to the nearest whole number:
Number_of_times = 3 times

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The MAD for this set of data is 0.8.

To find the MAD (Mean Absolute Deviation) for this set of data, we first need to find the mean of the ages:

Mean = (13 + 10 + 9 + 10 + 10 + 9 + 10 + 10 + 11 + 9) / 10 = 10.1

Next, we find the absolute deviation of each age from the mean:

|13 - 10.1| = 2.9

|10 - 10.1| = 0.1

|9 - 10.1| = 1.1

|10 - 10.1| = 0.1

|10 - 10.1| = 0.1

|9 - 10.1| = 1.1

|10 - 10.1| = 0.1

|10 - 10.1| = 0.1

|11 - 10.1| = 0.9

|9 - 10.1| = 1.1

Then, we find the average of these absolute deviations:

MAD = (2.9 + 0.1 + 1.1 + 0.1 + 0.1 + 1.1 + 0.1 + 0.1 + 0.9 + 1.1) / 10 = 0.8

Therefore,To find the MAD (Mean Absolute Deviation) for this set of data, we first need to find the mean of the ages:

Mean = (13 + 10 + 9 + 10 + 10 + 9 + 10 + 10 + 11 + 9) / 10 = 10.1

Next, we find the absolute deviation of each age from the mean:

|13 - 10.1| = 2.9

|10 - 10.1| = 0.1

|9 - 10.1| = 1.1

|10 - 10.1| = 0.1

|10 - 10.1| = 0.1

|9 - 10.1| = 1.1

|10 - 10.1| = 0.1

|10 - 10.1| = 0.1

|11 - 10.1| = 0.9

|9 - 10.1| = 1.1

Then, we find the average of these absolute deviations:

MAD = (2.9 + 0.1 + 1.1 + 0.1 + 0.1 + 1.1 + 0.1 + 0.1 + 0.9 + 1.1) / 10 = 0.8

Therefore, the MAD for this set of data is 0.8.

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Lesli ganará $750 de interés si presta $5000 a pagar en 3 años al 5% de interés simple anual.

Para calcular el interés que ganará Leslie en diferentes escenarios, consideraremos los siguientes casos:

A) Tasa simple anual del 5%:

El interés simple se calcula multiplicando el capital prestado por la tasa de interés y el tiempo en años.

Interés = Capital x Tasa x Tiempo

Interés = 5000 x 0.05 x 3 = $750

B) Tasa simple anual del 10%:

De manera similar al caso anterior, el interés se calcula como:

Interés = 5000 x 0.10 x 3 = $1500

C) Tasa compuesta anual del 5%:

En el caso de la tasa de interés compuesta, los intereses se acumulan en cada período. La fórmula para calcular el monto total es:

Monto = Capital x (1 + Tasa)^Tiempo

Monto = 5000 x (1 + 0.05)^3 = $5788.75

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Using the Empirical Rule.  The percentage of small business stock prices that are more than $81. 67 is 0.30%

To use the Empirical Rule, we need to assume that the distribution of small business stock prices is approximately normal.

First, we need to find the z-score for $81.67:

z = (81.67 - 54.82) / 8.95 = 2.99

Next, we can use the Empirical Rule to estimate the percentage of small business stock prices that are more than $81.67:

About 99.7% of the data falls within 3 standard deviations of the mean.

Therefore, about 0.3% of the data falls more than 3 standard deviations above the mean.

Since $81.67 is more than 3 standard deviations above the mean, we can estimate that the percentage of small business stock prices that are more than $81.67 is about 0.3%.

Rounded to the nearest hundredth, the estimated percentage is 0.30%.

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The x-coordinate of point T is 7. Thus, point T is (7, 3).

To solve for x, we will use the concept of slope. The slope between any two points on a line remains constant. Let's find the slope between points R(-5, -3) and S(-1, -1):

Slope (m) = (y2 - y1) / (x2 - x1)
m = (-1 - (-3)) / (-1 - (-5))
m = (2) / (4)
m = 1/2

Now, we will use the slope between points S(-1, -1) and T(x, 3):

m = (3 - (-1)) / (x - (-1))
1/2 = (4) / (x + 1)

Now, we will solve for x:

1/2 (x + 1) = 4
x + 1 = 8
x = 7

So, the x-coordinate of point T is 7. Thus, point T is (7, 3).

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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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The given differential equation is y"' + 4y = 0. To derive a recurrence relation, we assume that the solution has the form y = e^rx.

Substituting this in the differential equation, we get the characteristic equation r^3 + 4 = 0. Solving this, we get three roots r = -2i, 2i, 0.

So, the general solution is y = c1cos(2x) + c2sin(2x) + c3. Using the initial conditions y(0) = 0 and y'(0) = 1, we get c1 = 0 and c2 = 1/2.

Therefore, the solution is y = 1/2sin(2x) + c3.

Now, we can find the recurrence relation by writing c3 in terms of c2 and c1. We have c3 = y(0) - (1/2)sin(0) = 0. So, the recurrence relation is cn+2 = -4cn.

Using the initial conditions, we have c1 = 0 and c2 = 1/2. Therefore, the explicit formula for the coefficients is cn = (1/2)(-4)^n-2 for n ≥ 2.

Finally, the particular solution can be found by adding the general solution to the homogeneous solution. Since the roots are imaginary, the particular solution will have the form y = Acos(2x) + Bsin(2x).

Substituting this in the differential equation, we get A = 0 and B = -1/8.
So, the particular solution is y = -1/8sin(2x).

Therefore, the final solution in terms of familiar elementary functions is y = (1/2)sin(2x) - (1/8)sin(2x) = (3/8)sin(2x).

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Based on the results of Corey repeating his process 10 more times, a possible reason for this outcome with 3 green balls, 2 orange balls, and 5 purple balls could be that there is a higher probability of selecting a purple ball compared to the other colors.

Here's a step-by-step explanation:

1. Corey conducted an experiment where he repeated a process 10 times.
2. During these trials, he obtained the following results: 3 green balls, 2 orange balls, and 5 purple balls.
3. The distribution of colors suggests that there is a higher probability of selecting a purple ball (5/10) than a green ball (3/10) or an orange ball (2/10).
4. This outcome could be due to factors such as a larger number of purple balls in the pool from which Corey is selecting or some other bias in the process that increases the likelihood of selecting a purple ball.

In conclusion, the possible reason for the outcome with 3 green balls, 2 orange balls, and 5 purple balls is that there might be a higher probability of selecting a purple ball during Corey's repeated trials.

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If Tamekia and Marsha work for 22 hours, Marsha will earn $27.50 more than Tamekia.

To determine the amount of money each person can earn, we need to know the hourly rate and the number of hours worked. Tamekia estimates that she can earn between $6.00 and $6.25 per hour. Let's assume that she can earn $6.25 per hour. If Tamekia works for 22 hours, her total earnings will be:

Total earnings for Tamekia = Hourly rate x Number of hours worked = $6.25 x 22 = $137.50

Now let's look at Marsha's earnings. Marsha estimates that she can earn between $7.50 and $8.00 per hour. Let's assume that she can earn $7.50 per hour. If Marsha works for 22 hours, her total earnings will be:

Total earnings for Marsha = Hourly rate x Number of hours worked = $7.50 x 22 = $165.00

To determine how much more money Marsha will earn than Tamekia, we need to subtract Tamekia's total earnings from Marsha's total earnings:

Difference in earnings = Total earnings for Marsha - Total earnings for Tamekia = $165.00 - $137.50 = $27.50

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If p : q = 2/3 : 2 and p : r = 3/4 : 1/2 , the ratio of p : q : r in its simplest form is 32 : 27 : 24.

To calculate the ratio p : q : r, we need to first find the values of p, q, and r. We can use the given proportions to set up a system of equations and solve for the variables.

From the first proportion, we know that:

p/q = 2/3 : 2

We can simplify this by cross-multiplying:

p = (2/3) * 2q

p = (4/3)q

From the second proportion, we know that:

p/r = 3/4 : 1/2

Again, we can cross-multiply and simplify:

p = (3/4) * r/(1/2)

p = (3/2)r

Now we have two equations for p in terms of q and r. We can substitute these into each other and solve for q and r:

(4/3)q = (3/2)r

r/q = (8/9)

q/r = (9/8)

Now we have the ratios of r to q and q to r. We can use these to find the ratio of p, q, and r:

p : q : r = p : q * (9/8) : r * (8/9)

Substituting the values we found for p in terms of q and r:

p : q : r = (4/3)q : q * (9/8) : r * (8/9)

Simplifying:

p : q : r = 32 : 27 : 24

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To evaluate the definite integral of the given function, ∫ (t^5 - 2t^2)/t^4 dt, follow these steps:
1. Simplify the integrand: Divide each term by t^4.
  (t^5/t^4) - (2t^2/t^4) = t - 2t^(-2)

2. Integrate each term with respect to t.
  ∫(t dt) - ∫(2t^(-2) dt) = (1/2)t^2 + 2∫(t^(-2) dt)

3. Apply the power rule to the remaining integral.
  (1/2)t^2 + 2(∫t^(-2+1) dt) = (1/2)t^2 + 2(∫t^(-1) dt)

4. Integrate t^(-1) with respect to t.
  (1/2)t^2 + 2(ln|t|)

Now, since we need to evaluate the definite integral, we should have the limits of integration. Let's assume the limits of integration are a and b. Then, apply the Fundamental Theorem of Calculus:

[(1/2)b^2 + 2(ln|b|)] - [(1/2)a^2 + 2(ln|a|)]

This expression gives the value of the definite integral for the given function within the limits a and b.

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The completed table based on the given pattern is as follows: Shape 1: 4 matchsticks, Shape 2: 7 matchsticks, Shape 3: 10 matchsticks, Shape 4: 13 matchsticks, Shape 5: 24 matchsticks, Shape 6: 82 matchsticks.

To complete the table based on the given pattern, we need to identify the rule that determines the number of matchsticks for each shape.

Looking at the provided information, we can observe that the first four shapes follow a consistent rule, while the last two shapes seem to deviate from that rule.

For the first four shapes:

Shape 1: 4 matchsticks

Shape 2: 7 matchsticks (Shape 1 + 3 matchsticks)

Shape 3: 10 matchsticks (Shape 2 + 3 matchsticks)

Shape 4: 13 matchsticks (Shape 3 + 3 matchsticks)

Based on this pattern, it appears that each shape adds three additional matchsticks compared to the previous shape.

Now, let's analyze the last two shapes:

Shape 6: 43 matchsticks

Shape 7: 82 matchsticks

From Shape 4 to Shape 6, there is an increase of 3 matchsticks as expected.

However, from Shape 6 to Shape 7, there is an unexpected increase of 39 matchsticks.

Since the given information does not provide a clear pattern or rule for the last two shapes, we cannot accurately determine the number of matchsticks for those shapes.

Therefore, we can complete the table as follows:

Shape 1: 4 matchsticks

Shape 2: 7 matchsticks

Shape 3: 10 matchsticks

Shape 4: 13 matchsticks

Shape 5: (Unknown)

Shape 6: (Unknown).

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The trigonometric ratios of angle C are sin C = 15/17, cos C = 8/17, and tan C = 15/8.

Since triangle ABC is a right triangle with a right angle at B, and we know AC = 17 units (hypotenuse) and BC = 8 units (adjacent side to angle C), we can use the Pythagorean theorem to find the length of the remaining side, AB (opposite side to angle C).

The Pythagorean theorem states: AB² + BC² = AC²

Plugging in the values we know:
AB² + 8² = 17²
AB² + 64 = 289

To find AB:
AB² = 289 - 64 = 225
AB = √225 = 15 units

Now we can determine the trigonometric ratios of angle C:

1. sine (sin C) = opposite/hypotenuse = AB/AC = 15/17
2. cosine (cos C) = adjacent/hypotenuse = BC/AC = 8/17
3. tangent (tan C) = opposite/adjacent = AB/BC = 15/8

So the trigonometric ratios of angle C are:
sin C = 15/17, cos C = 8/17, and tan C = 15/8.

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Your best friend is charging you an annual interest rate of 4% for lending you ₹300 for nine months with a quarterly interest rate of 1%.

The amount a lender charges a borrower for the use of assets, such as money, consumer goods, or physical assets, is known as an interest rate. It is a fraction of the loan's principal, which is the amount borrowed to cover the cost of the purchase or the deposit made with a bank or other financial institution.

If your best friend is charging you an extra ₹3 for every three months that pass before you return the money, then after nine months, you will owe her an extra ₹9 in addition to the original ₹300.

So the total amount you must pay her if you return the ₹300 after nine months would be ₹309.

To calculate the annual rate of interest she is charging you, we can use the formula:

Annual Interest Rate = (Total Interest / Principal) x (12 / Number of Months)

Where the Principal is the original amount borrowed (₹300), the Total Interest is the extra amount you owe her (₹9), and the Number of Months is the time period for which you borrowed the money (9 months).

Plugging in the values, we get:

Annual Interest Rate = (9 / 300) x (12 / 9) = 0.04 or 4%

So, your best friend is charging you an annual interest rate of 4% for lending you ₹300 for nine months with a quarterly interest rate of 1%.

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The complete question is:

You ask your best friend to lend you ₹300 to buy your favourite toy. She says she can lend you the money only if you give her an extra ₹3 for every three months that pass before you return it. What is the total amount you must pay her if you return it after nine months? What is the annual rate of interest she is charging you?

The range of the data set is 19 hours. None of the options are correct.

The given data set represents the number of hours Raymond spent practicing the piano each week for several weeks: 24, 8, 6, and 5 hours. To calculate the range, follow these steps:

1. Identify the highest and lowest values in the data set.
  - The highest value is 24 hours.
  - The lowest value is 5 hours.

2. Subtract the lowest value from the highest value to find the range.
  - Range = Highest value - Lowest value
  - Range = 24 hours - 5 hours
  - Range = 19 hours

This means that there is a 19-hour difference between the maximum and minimum number of hours Raymond spent practicing the piano each week. None of the options (a. 8 hours, b. 2 hours, c. 5 hours, and 6 hours) provided matches the correct answer, which is 19 hours.

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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.

To dy/dx when x = - 1 and dx/dt =5. dy/dt =

dy/dt = -35.

To find dy/dt, first we need to find dy/dx. Given y = 5x^2 + 3x, we can differentiate y with respect to x:

[tex]dy/dx = d(5x^2 + 3x)/dx = 10x + 3[/tex]
Now, we need to find dy/dx when x = -1:

[tex]dy/dx(-1) = 10(-1) + 3 = -10 + 3 = -7[/tex]
We are given that dx/dt = 5. To find dy/dt, we use the chain rule:

[tex]dy/dt = dy/dx * dx/dt[/tex]

Substitute the values we found:

dy/dt = (-7) * (5) = -35

So, dy/dt = -35.

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Answer:

B

Step-by-step explanation:

Since the graph is facing down, there will be a negative sign.

In parenthesis it says (x + 1) which means you move one unit left

On the outside it says +2 which means you move the graph 2 units up

By using concept of interior angle we find the value of X is -7.14 degrees.

The above problem involves finding the value of x in a triangle with two known angles measuring 78° and 152°.

The sum of the interior angles of any triangle is always 180°, so we can use this fact to set up an equation involving the third angle, which is given as 7x +3 degrees.

To solve for x, we first simplify the equation by combining the known angles:

78° + 152° + (7x + 3)° = 180°

Next, we can simplify by adding the two known angles:

230° + 7x° = 180°

This simplifies to:

7x° = -50°

Finally, we can solve for x by dividing both sides by 7:

x =  [tex]\frac{-50^\circ}{7}$$[/tex]

Therefore, x is approximately -7.14 degrees.

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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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a) The demand function is 134.33e^-0.693p

b) At P = $3, we have elasticity is  0.83, at P = $4, we have elasticity is  1.05, at P = $5, we have elasticity is 1.34.

c) We should sell the cheese at a price of $3.84 per pound to maximize monthly revenue.

d) We should sell the cheese at a price of $4.22 per pound to generate the highest revenue within the timeframe of one month.

a) To construct a demand function of the form q = Ae^-bp, we can use the sales figures for the prices $3 and $5 per pound. First, we calculate the values of A and b:

A = q/p = 403/3 ≈ 134.33

b = ln(q/Ap) / p = ln(403/134.33) / (3-5) ≈ 0.693

Using these values, the demand function becomes:

q = 134.33e^-0.693p

b) To find the price elasticity of demand at each of the prices listed, we can use the formula:

E = (dq/dp) * (p/q)

At P = $3, we have:

E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 3) / 403 ≈ 0.83

At P = $4, we have:

E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 4) / 284 ≈ 1.05

At P = $5, we have:

E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 5) / 225 ≈ 1.34

c) To find the price that will maximize monthly revenue, we can use the formula:

p = (1/b) * ln(A/b)

Plugging in the values of A and b that we calculated earlier, we get:

p = (1/0.693) * ln(134.33/0.693) ≈ $3.84

d) If we only have 200 pounds of cheese and it will spoil in one month, we need to sell it at a price that will generate the highest revenue within that timeframe. To do this, we can use the formula:

R = pq

where R is the revenue, p is the price per pound, and q is the quantity sold. We can express q in terms of p using our demand function:

q = 134.33e^-0.693p

Substituting this into the revenue equation, we get:

R = p * 134.33e^-0.693p

To find the price that will maximize revenue, we can take the derivative of R with respect to p and set it equal to zero:

dR/dp = 134.33e^-0.693p - 93.13pe^-0.693p = 0

Solving this equation numerically, we get:

p ≈ $4.22

This price is different from the price calculated in part (c) because we have a limited quantity of cheese that will spoil, so we need to balance the price and quantity sold to maximize revenue within the given timeframe.

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