A block of wood measures 6.5 inches by 1.5 inches by 8 inches. What is the volume of the block of wood?
Type your answer with cubic inches.

A. 52 = 8n + 4, B. 4(2n + 1) = 52 and C. 4 = 52 – 8n are the equivalent equations.

Simplifying the equations, we have

A. 52 = 8n + 4

8n = 48

n = 6

To see this, first simplify equation B:

4(2n + 1) = 52

8n + 4 = 52

8n = 48

n = 6

C. 4 = 52 – 8n

8n = 48

n = 6

Then simplify equation D:

4n = 48

n = 12

As you can see, equations A, B and C are equivalent and both simplify to n = 6.

Therefore, the correct answers are A, B and C.

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The radius of gyration of the plate about the x-axis is [tex]6 \sqrt{6} / 5[/tex] units.

To find the radius of gyration of a plate covering the region bounded by [tex]y = x^2[/tex], x = 6, and the x-axis with respect to the x-axis, we need to use the formula:

[tex]k_x = \sqrt{(I_x / A)}[/tex]

where [tex]k_x[/tex] is the radius of gyration, [tex]I_x[/tex] is the moment of inertia of the plate about the x-axis, and A is the area of the plate.

We can calculate the area A of the plate as follows:

[tex]A = \int\limits^6_0 { x^2}\, dx\\= [x^3/3]\ from\ 0\ to\ 6\\= 72[/tex]

To find the moment of inertia [tex]I_x[/tex], we can use the formula:

[tex]I_x = \int\ {y^2} \, dA[/tex]

where y is the perpendicular distance of an element of area [tex]dA[/tex] from the x-axis. We can express y in terms of x as y = x². Therefore, we have:

[tex]dA = y dx = x^2 dx\\I_x = \int\limits^6_0 { x^2 (x^2)} dx\\= \int\limits^6_0 {x^4}\, dx\\= [x^5/5]\ from\ 0\ to\ 6\\= 6^5/5[/tex]

Substituting these values into the formula for [tex]k_x[/tex], we get:

[tex]k_x = \sqrt{(I_x / A)}\\= \sqrt{((6^5/5) / 72)}\\= \sqrt{(6^3 / 5)}\\= 6 \sqrt{6} / 5[/tex]

Therefore, the radius of gyration of the plate about the x-axis is [tex]6 \sqrt{6}/ 5[/tex] units.

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The height of the triangle is given as follows:

[tex]h = 2\sqrt{3}[/tex] cm.

The area of the triangle is given as follows:

[tex]A = 4\sqrt{3}[/tex] cm².

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

Considering an equilateral triangle, in which all the side lengths are of 4, we have a right triangle in which:

Hence the height is obtained as follows:

h² + 2² = 4²

h² = 12

[tex]h = \sqrt{3 \times 4}[/tex]

[tex]h = 2\sqrt{3}[/tex] cm

The area of a triangle is given as half the multiplication of the base and of the height, hence:

[tex]A = 0.5 \times 4 \times 2 \sqrt{3}[/tex]

[tex]A = 4\sqrt{3}[/tex] cm².

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Answer: $193.42

Step-by-step explanation:

Based on the given designs, the cost to make the triangular and circular street signs would be $68.60 and $124.42 respectively, making the total cost of making both signs $193.02.

To calculate the cost of making the two street signs, we need to first find the area of each sign. The area of a triangle is given by the formula 1/2 x base x height. So, for the triangular street sign, the area would be 1/2 x 3 x 2.6 = 3.9 square feet.

The area of a circle is given by the formula π x radius². So, for the circular street sign, the area would be π x (1.5)² = 7.065 square feet.

Now that we have the areas of both signs, we can calculate the total cost of the material needed. The cost per square foot of material is $17.64, so we need to multiply this by the total area of the signs.

For the triangular sign, the cost would be 3.9 x $17.64 = $68.60.

For the circular sign, the cost would be 7.065 x $17.64 = $124.42.

Therefore, the total cost to make both signs would be $68.60 + $124.42 = $193.02.

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The mass of iron III sulfate is  300 g.

If 1 mole of iron III sulfate is produced when 3 moles of hydrogen is produced

x moles of iron III sulfate is produced when 2.25 moles of hydrogen is produced

x = 1 * 2.25/3

= 0.75 moles

Mass of the iron III sulfate = 0.75 moles * 400 g/mol

= 300 g

Number of moles of iron = 85 g/56 g/mol

= 1.5 moles

If 2 moles of iron produces 1 mole of   iron III sulfate

1.5 moles of iron produces 1.5 * 1/2

= 0.75 moles

Mass of the  iron III sulfate = 0.75 moles * 400 g/mol

= 300 g

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The graph of constant of proportionality of y = 3.75x is attached

The constant of proportionality is a term that indicates a reciprocal relationship between two variables, in which the change of one affects the other similarly.

When x and y are directly linked in this way, the following equation can be used to calculate how they operate together:

y = kx,

where

k serves as the aforementioned constant.

In the problem k = 3.75

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The dealer should have sold the article for 103,500 naira to make a profit of 15%.

Let C be the cost price of the composition.  According to the problem, the dealer  vended the composition at a loss of 10, so he  entered 90 of the cost price.  thus, 90 of C is equal to 81,000 naira.

C =  81,000

C =  81,000/0.9

 C =  90,000  

So, the cost price of the composition is 90,000 naira.

  Now, let's find out the selling price  needed to make a profit of 15   Let S be the  needed selling price to make a profit of 15.  We know that profit chance is equal to( profit/ cost price) × 100.

Thus,(15/100) × 90,000 =

S- 90,000 , 500

=  S- 90,000  

S =  90,000 13,500

S =  103,500

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Answer:

5,005

Step-by-step explanation:

This is a combination problem. The formula for combination is:

nCr = n! / (r!(n-r)!)

Where n is the total number of items, and r is the number of items to be selected.

Using this formula, we can calculate the number of ways to choose 9 varieties from 15:

15C9 = 15! / (9!(15-9)!) = 5005

Therefore, there are 5,005 ways to choose 9 varieties from 15 varieties of pansies.

She is incorrect when she said that her rate increased each hour from 8 to 12 to 16 miles per hour. Her jogging rate was actually a steady 5 miles per hour.

You asked if Christy is correct when she said that her rate increased each hour, from 8 to 12 to 16 miles per hour. Let's analyze the data given in the table and see if she's right.

The table shows the distance Christy jogged and the corresponding time:
- 10 miles in 2 hours
- 15 miles in 3 hours
- 20 miles in 4 hours

To find her jogging rate for each time period, we need to divide the distance by the time.

1. For the first 2 hours:
Jogging rate = distance / time = 10 miles / 2 hours = 5 miles per hour

2. For the first 3 hours:
Jogging rate = distance / time = 15 miles / 3 hours = 5 miles per hour

3. For the first 4 hours:
Jogging rate = distance / time = 20 miles / 4 hours = 5 miles per hour

Christy's jogging rate remained constant at 5 miles per hour throughout her run. Therefore, she is incorrect when she said that her rate increased each hour from 8 to 12 to 16 miles per hour. Her jogging rate was actually a steady 5 miles per hour.

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You can fill 4 five liter gas tanks from a full 20 liter gas can.

To determine how many 5-liter gas tanks can be filled from a full 20-liter gas can, you would divide the total capacity of the gas can by the capacity of the individual gas tanks.

Step 1: Identify the total capacity of the gas can (20 liters) and the capacity of each gas tank (5 liters).
Step 2: Divide the total capacity by the individual tank capacity (20 liters / 5 liters).

Your answer: You can fill 4 (20 liters / 5 liters = 4) 5-liter gas tanks from a full 20-liter gas can.

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To find the elasticity of demand for the function q = D(x) = 460 - x, we can use the formula:

Elasticity = (% change in quantity demanded) / (% change in price)

a) Since the demand function given does not include a price variable, we can assume that price is constant. Therefore, the elasticity of demand for this function is constant and equal to -1.

b) To find the elasticity at x = 103, we need to calculate the percentage change in quantity demanded when x increases from 103 to 104.

At x = 103, quantity demanded is q = D(103) = 460 - 103 = 357.

At x = 104, quantity demanded is q = D(104) = 460 - 104 = 356.

The percentage change in quantity demanded is:

(% change in quantity demanded) = [(new quantity - old quantity) / old quantity] x 100

= [(356 - 357) / 357] x 100 = -0.28%

Since the elasticity of demand is -1, we can say that demand is inelastic at x = 103.

c) To find the elasticity at x = 205, we need to calculate the percentage change in quantity demanded when x increases from 205 to 206.

At x = 205, quantity demanded is q = D(205) = 460 - 205 = 255.

At x = 206, quantity demanded is q = D(206) = 460 - 206 = 254.

The percentage change in quantity demanded is:

(% change in quantity demanded) = [(new quantity - old quantity) / old quantity] x 100

= [(254 - 255) / 255] x 100 = -0.39%

Since the elasticity of demand is -1, we can say that demand is inelastic at x = 205.
Hi! I'd be happy to help you with your question on elasticity of demand.

a) To find the elasticity, we first need the formula for price elasticity of demand (PED), which is:

PED = (% change in quantity demanded) / (% change in price)

Here, we have the demand function D(x) = 460 - x, where x is the price.

b) To find the elasticity at x = 103, we first need to calculate the quantity demanded, which is:

q = D(103) = 460 - 103 = 357

Now, we'll find the derivative of the demand function with respect to price:

dq/dx = -1

Next, we'll use the formula for PED:

PED = (dq/dx * x) / q = (-1 * 103) / 357 = -103/357 ≈ -0.289

Since the absolute value of PED is less than 1, demand is inelastic at x = 103.

c) To find the elasticity at x = 205, we'll follow the same steps:

q = D(205) = 460 - 205 = 255

PED = (dq/dx * x) / q = (-1 * 205) / 255 ≈ -0.804

Again, the absolute value of PED is less than 1, so demand is inelastic at x = 205.

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P(E1) + P(E2) - P(E1 ∩ E2) is the probability that at least one of the two events occurs in any trial of the experiment.

The correct answer is D. P(E1) + P(E2) - P(E1 ∩ E2).

The probability that at least one of two events occurs can be calculated using the principle of inclusion-exclusion.

The formula for this is:

P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)

Where:

P(E1) is the probability of event E1 occurring.

P(E2) is the probability of event E2 occurring.

P(E1 and E2) is the probability of both events E1 and E2 occurring simultaneously.

This formula represents the probability that at least one of the two events (E1 or E2) occurs in any trial of the experiment.

It's derived using the principle of inclusion-exclusion.

P(E1) represents the probability of event E1 occurring.

P(E2) represents the probability of event E2 occurring.

P(E1 ∩ E2) represents the probability of both events E1 and E2 occurring simultaneously.

By adding the probabilities of each individual event and then subtracting the probability of their intersection, you're accounting for the possibility of double-counting the intersection when adding the probabilities of the individual events.

So, the formula accurately captures the probability of at least one of the two events occurring.

Hence, P(E1) + P(E2) - P(E1 ∩ E2) is the probability that at least one of the two events occurs in any trial of the experiment.

The correct answer is D. P(E1) + P(E2) - P(E1 ∩ E2).

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complete question:

Two events, E1 and E2, are defined for a random experiment. What is the probability that at least one of the two events occurs in any trial of the experiment?

A.

P(E1) + P(E2) − 2P(E1 ∩ E2)

B.

P(E1) + P(E2) + P(E1 ∩ E2)

C.

P(E1) − P(E2) − P(E1 ∩ E2)

D.

P(E1) + P(E2) − P(E1 ∩ E2)

To translate one-step equations, you need to understand the language of algebra.

Algebraic expressions involve variables, numbers, and operations such as addition, subtraction, multiplication, and division. One-step equations require only one operation to isolate the variable, making them easy to solve.

To write an equation to represent the statement "15 is 9 more than j," you can use the equation 15 = j + 9. This equation says that 15 is equal to j plus 9. To solve for j, you need to isolate j on one side of the equation by subtracting 9 from both sides. This gives you the equation j = 6.

To solve the equation j % 3 = c, you need to understand the modulus operator, which gives you the remainder when two numbers are divided. In this case, j % 3 means the remainder when j is divided by 3. To solve for j, you need to multiply both sides of the equation by 3, which gives you the equation j = 3c.

In summary, to translate one-step equations, you need to understand the language of algebra and the operations involved. To solve for variables, you need to isolate them on one side of the equation. And to solve equations involving the modulus operator, you need to understand how it works and how to apply it to solve for variables.

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Bristol is the cheapest city and Paul can drop from his plans to stay under budget.

To determine which city Paul can drop and still stay under budget, we need to find the total cost of visiting all four cities and compare it to Paul's budget of £300.

We do not have information about the cost of sightseeing in each city, so we will assume that the cost is the same for each city. In that case, the cost of visiting all four cities is:

Cost of visiting all cities = Cost of visiting one city x 4

Let's call the cost of visiting one city "x". Then:

Cost of visiting all cities = x * 4

To stay under budget, the cost of visiting all cities must be less than or equal to £300. So we can write the following inequality:

Simplifying this inequality, we get:

x ≤ £75

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The answer is (c) Bristol, as it has the highest cost among the remaining cities and dropping it would leave Paul with 224 pounds, which is still within his budget.

A budget is whenever one plans on how to spend an estimated income. All the income should be considered as well as all the expenses. In other words, it is an expending plan.

The total cost of visiting all cities is:

76 + 66 + 91 + 60 + 72 = 365

To stay under budget, Paul must drop at least one city. To determine the cheapest city to drop, we can calculate the cost of the trip to each city if visited individually and compare it to the remaining budget:

- Bristol: 76 pounds

 Remaining budget: 300 - 76 = 224 pounds

- Leicester: 66 pounds

 Remaining budget: 300 - 66 = 234 pounds

- Glasgow: 91 pounds

 Remaining budget: 300 - 91 = 209 pounds

- Leeds: 60 pounds

 Remaining budget: 300 - 60 = 240 pounds

- Belfast: 72 pounds

 Remaining budget: 300 - 72 = 228 pounds

The city that Paul can drop while staying under budget is the one with the lowest cost.

Therefore, the answer is (c) Bristol, as it has the highest cost among the remaining cities and dropping it would leave Paul with 224 pounds, which is still within his budget.

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The absolute value of -8 1/2 is less than the absolute value of -9 1/2.

To compare the absolute values of -8 1/2 and -9 1/2, follow these steps:
1. Convert the mixed numbers to improper fractions:
-8 1/2 = -17/2
-9 1/2 = -19/2

2. Find the absolute values of both numbers:
|-17/2| = 17/2
|-19/2| = 19/2

3. Compare the absolute values and choose the correct symbol:
17/2 < 19/2
So, the statement is: The absolute value of -8 1/2 is less than the absolute value of -9 1/2.

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(a) Linus's score for Test 1 is 80. (b) The percentage increase in his test score from Test 2 to Test 3 is 4.3%.

(a) Let's denote Linus's score in Test 1 as "x." Since he received 12 more marks in Test 2, his score for Test 2 is "x + 12." The 15% improvement means that (x + 12) is 115% of x:

x + 12 = 1.15x

Now, we can solve for x:

12 = 0.15x

x = 12 / 0.15

x = 80

So, Linus's score in Test 1 was 80.

(b) Linus made a 4-mark improvement in Test 3, so his score was (x + 12) + 4, which is (80 + 12) + 4 = 96. To find the percentage increase from Test 2 to Test 3, we can use the formula:

Percentage increase = ((New score - Old score) / Old score) * 100

Percentage increase = ((96 - 92) / 92) * 100

Percentage increase = (4 / 92) * 100

Percentage increase ≈ 4.35

The percentage increase in his test score from Test 2 to Test 3 is approximately 4.3% (correct to 1 decimal place).

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The count of duration that is needed for Dr. Aghedo's money to be deposited is 22.3 years, under the condition that if interest is compounded continuously at a rate of 3. 1

The derived formula for doubling time with continuous compounding is applied to evaluate the length of time it takes to double the money in an account or investment that has continuous compounding. The formula is
Doubling time = ln 2 / r

Here,
r =annual interest rate as a decimal.

For the required case, the interest rate is 3.1% that can be written as  0.031 in the form of decimal. Then the doubling time will be
Doubling time = ln 2 / 0.031
≈ 22.3 years

Then, it should take approximately 22.3 years for Dr. Aghedo's money to double if interest is compounded continuously at a rate of 3.1%.
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Answer:

Step-by-step explanation:

I can provide you with general information about radical functions and their graphs, but I cannot design a bridge for you.

In order to design a bridge that will safely cross the ravine, you would need to take into account a wide range of factors, including the length and width of the ravine, the types of materials that can be used to construct the bridge, the weight and size of the vehicles that will be crossing the bridge, and the weather and environmental conditions in the area. This would likely require the expertise of a civil engineer or other trained professional.

Regarding radical functions, they are functions that involve a radical symbol (such as a square root) in their equation. The graph of a radical function is typically a curve that starts at the point (0,0) and moves upwards and to the right. The shape of the curve will depend on the specific radical function and the values of its parameters.

To solve a radical function, you would typically isolate the radical term on one side of the equation and then square both sides of the equation to eliminate the radical. However, it is important to be careful when squaring both sides, as this can introduce extraneous solutions that do not satisfy the original equation.

Answer:

The system of equations to describe the situation is:

* $10x + 54 = 11y$

* $y = 11x$

To solve using substitution, we can substitute the second equation into the first equation. This gives us:

* $10x + 54 = 11(11x)$

* $10x + 54 = 121x$

* $-11x = -54$

* $x = 5$

We can then substitute this value of $x$ into the second equation to solve for $y$. This gives us:

* $y = 11(5)$

* $y = 55$

Therefore, the student council will need to sell 55 T-shirts to cover their costs. The costs will be $54 + $10(55) = $604.

Here is the solution in a table format:

| Variable | Value |

|---|---|

| $x$ | 5 |

| $y$ | 55 |

| Cost | $604 |

I hope this helps!

Step-by-step explanation:

The  function f(x)=500[tex]x^2[/tex] +10,000  can  be  used  to  model  the  food  vendor's  annual  revenue. The  food  vendor  would expect  to  bring  in  $ 60,000  in  annual  revenue by 10 years from now.

To find out when the mobile food vendor should expect to bring in $60,000 in annual revenue using the function f(x) = 500x^2 + 10,000, follow these steps:

1. Set the function equal to 60,000: 500[tex]x^2[/tex]+ 10,000 = 60,000
2. Subtract 60,000 from both sides of the equation: 500[tex]x^2[/tex]+ 10,000 - 60,000 = 0
3. Simplify the equation: 500[tex]x^2[/tex]- 50,000 = 0
4. Divide both sides by 500:[tex]x^2[/tex] - 100 = 0
5. Add 100 to both sides: [tex]x^2[/tex] = 100
6. Find the square root of both sides: x = ±10

Since it doesn't make sense to have a negative number of years, the food vendor should expect to bring in $60,000 in annual revenue 10 years from now.

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Answer:

C.

Step-by-step explanation:

3 = 1.5 * 2

9 = 1.5 * 6

15 = 1.5 * 10

33 = 1.5 * 22

The given values for uo and u3 are uo = 1 and u3 = 21. We can use the recurrence relation Un+2 = 2 * Un+1+1* Un to find the remaining terms:

U1 = U3 - 2U2 - 1*U0
U1 = 21 - 2U2 - 1*1
U1 = 20 - 2U2

U2 = U1 - 2U0 + 1*U0
U2 = 20 - 2U0 + 1*1
U2 = 19 - 2U0

Therefore, the first four terms are: 1, 19, -17, -53

So, the answer is: 1, 19, -17, -53.

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If a number cube is rolled twice, then the probability of getting a six on the first role and then a number less than 5 on the second role is 1/9.

To find the probability of getting a six on the first roll and a number less than 5 on the second roll, we will multiply the individual probabilities of each event.

A number cube has 6 faces, so the probability of rolling a six is 1/6.

For the second roll, there are 4 numbers less than 5 (1, 2, 3, and 4), so the probability of rolling a number less than 5 is 4/6 or 2/3.

To find the combined probability, simply multiply the two probabilities: (1/6) × (2/3) = 2/18 = 1/9.

So, the probability of getting a six on the first roll and a number less than 5 on the second roll is 1/9.

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There are 5 combinations with one flavor, 10 combinations with two flavors, and 10 combinations with three flavors, resulting in 25 possible combinations in total.

a. Only one flavor is there.

b. Buko can be paired with 4 other flavors and Ube can be paired with 3 remaining flavors.

c. 10 combinations of three flavors.

a. There is only one flavor: Since the box contains only one flavor, there are 5 possible combinations (Buko, Ube, Pineapple, Yema, and Mango).

b. There are two flavors:
- Buko can be paired with 4 other flavors (Ube, Pineapple, Yema, Mango).
- Ube can be paired with 3 remaining flavors (Pineapple, Yema, Mango).
- Pineapple can be paired with 2 remaining flavors (Yema, Mango).
- Yema can be paired with 1 remaining flavor (Mango).
In total, there are 10 combinations of two flavors.

c. There are three flavors:
- There are 5 flavors in total, and we want to choose 3. We can use the formula for combinations: C(n, k) = n! / (k!(n-k)!), where n is the total number of flavors and k is the number of flavors to choose.
- C(5, 3) = 5! / (3!(5-3)!) = 10 combinations of three flavors.

So, there are 5 combinations with one flavor, 10 combinations with two flavors, and 10 combinations with three flavors, resulting in 25 possible combinations in total.

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To answer question a, we use the formula:

time taken = (number of tyres to fit x time taken to fit one tyre) / number of tyres fitted at once

In this case, Liam takes 124 minutes to fit 4 tyres to a lorry. To find out how long it would take him to fit 6 tyres, we plug in the values:

time taken = (6 x 124) / 4
time taken = 186 minutes

So it would take Liam 186 minutes to fit 6 tyres to a lorry.

For question b, we know that Liam takes 124 minutes to fit 4 tyres, so he takes 31 minutes to fit 1 tyre. If he works for 93 minutes, we can find out how many tyres he can fit:

number of tyres = time taken / time taken to fit one tyre
number of tyres = 93 / 31
number of tyres = 3

So Liam can fit 3 tyres in 93 minutes.

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The vertex of the function is -6.

The domain of the function is {real numbers}.

The range of the function is y ≥ -6.

The range and domain of the function is calculated as follows;

y = x² - 2x - 5

From the graph of the function, the range of the function includes y values;

range = y ≥ -6

From the graph of the function, the domain of the function includes x values;

domain = real numbers.

The vertex of the graph is -6.

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To solve exercise 3 on the first page of the handout, we need to first isolate y in the given equation 2.23 - 3y = 8, which gives us y = -1.59.

To find y", we need to differentiate both sides of the equation with respect to x twice. The first derivative gives us:
-3(dy/dx) = 0
Simplifying, we get dy/dx = 0.
Differentiating again, we get:
-3d^2y/dx^2) = 0
Simplifying, we get d^2y/dx^2 = 0.

Therefore, the equation we obtain after differentiating both sides of the given equation with respect to x is d^2y/dx^2 = 0, which is the second derivative of y with respect to x.
To solve the equation given on the first page of the handout, 2.23 - 3y = 8, first isolate y:
1. Subtract 2.23 from both sides: -3y = 5.77
2. Divide both sides by -3: y = -5.77/3
Your answer: y = -5.77/3

The original equation doesn't have any x terms, so differentiation with respect to x is not applicable in this case.

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The average number of books bought per year by customers in the survey is approximately 4.85 books.

To find the average number of books bought per year, we need to calculate the mean of the data set. We can do this by using the formula:

Average = (Sum of all data points) / (Number of data points)

However, we do not have the actual number of data points. Instead, we have percentages. Therefore, we need to convert the percentages into actual numbers.

Out of 100 customers surveyed:

30% bought 3 books, which is equal to 30/100 x 100 = 30 customers

25% bought 5 books, which is equal to 25/100 x 100 = 25 customers

45% bought 6 books, which is equal to 45/100 x 100 = 45 customers

Now, we can calculate the average number of books bought per year using the formula mentioned earlier:

Average = (30 x 3) + (25 x 5) + (45 x 6) / (30 + 25 + 45)

Simplifying the above equation, we get:

Average = (90 + 125 + 270) / 100

Therefore, the average number of books bought per year is:

Average = 485/100

Average = 4.85 books per year (rounded to two decimal places)

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