A school rewards students for good work by giving them merit points. James has x merit points.
Sarah has three times as many merit points than James.
Robert has 59 fewer merit points than James. Each merit point is worth 3 pence.
All three of the students have a total of £11.88 Work out how many merit points each student has.

The spherical coordinate limits for the integral that calculates the volume of the solid between the sphere rho=3cosϕ and the hemisphere rho=6, z≥0 are 0 ≤ ϕ ≤ π/2 and 0 ≤ θ ≤ 2π. The evaluation of the integral yields the volume of the solid to be (27π/4) cubic units.

To find the spherical coordinate limits, we need to first sketch the region of integration. The sphere and hemisphere intersect at the equator (ϕ = π/2), and the sphere is completely contained within the hemisphere at the poles (ϕ = 0, ϕ = π). Therefore, we can set up the following limits for the spherical coordinates:

0 ≤ ϕ ≤ π/2 (hemisphere region)

0 ≤ θ ≤ 2π (full circle around z-axis)

3cos(ϕ) ≤ ρ ≤ 6 (region between sphere and hemisphere)

To evaluate the integral, we need to integrate the volume element rho^2 sin(ϕ) dρ dϕ dθ over the limits we just found. So the integral is:

∭V rho^2 sin(ϕ) dρ dϕ dθ

= ∫0^π/2 ∫0^2π ∫3cos(ϕ)^6 ρ^2 sin(ϕ) dρ dθ dϕ

= ∫0^π/2 ∫0^2π [1/3 ρ^3 sin(ϕ)]3cos(ϕ)^6 dθ dϕ

= ∫0^π/2 [2π/3 sin(ϕ)]3cos(ϕ)^6 dϕ

= (2π/3) ∫0^π/2 sin(ϕ)3cos(ϕ)^6 dϕ

Evaluating this integral requires a trigonometric substitution. Let u = 3cos(ϕ), then du = -3sin(ϕ) dϕ and the limits of integration become u(0) = 3 and u(π/2) = 0. Substituting in the integral, we get:

(2π/3) ∫3^0 (-1/3) u^6 du

= (2π/9) [u^7]3^0

= (2π/9) (3^7)

= 5103π/9

Simplifying, we get:

V = 567π

Therefore, the volume of the solid is 567π cubic units.

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The initial value of the function is 1, and the rate is 0.25

The equation of the exponential function is given as:

\(y = 0.25^x\)

An exponential function may represent growth or decay, and it is represented as:

\(y = ab^x\)

Where:

So, by comparison:

\(a =1\) and \(b = 0.25\)

Hence, the initial value of the function is 1, and the rate is 0.25

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How far both planes travelled is an illustration of distance

Both planes are 510 km apart at 2 pm

The rates of the two planes are given as:

Plane A = 300km/h

Plane B = 45km/h

Calculate the difference (r) in their rates

r = 300km/h - 45km/h

This gives

r = 255km/h

At 2 pm, both planes have travelled for 2 hours.

So, the distance between them is:

d = 255 * 2

d = 510km

Hence, both planes are 510 km apart at 2 pm

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Answer:

Dispensed

Step-by-step explanation:

The zeroes of the function \(f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45\) are x = 1, x = 3, and x = -5.

To find all zeroes of the function \(f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45\), follow these steps:

Step 1: Look for any rational roots using the Rational Root Theorem. This theorem states that if there is a rational root of the form p/q, then p must be a factor of the constant term (in this case, -45), and q must be a factor of the leading coefficient (in this case, 1).

Step 2: Test possible rational roots by plugging them into the function f(x). If f(x) = 0, then that value of x is a root. In this case, since the leading coefficient is 1, the possible rational roots are the factors of -45, which are ±1, ±3, ±5, ±9, ±15, and ±45.

Step 3: After testing each possible rational root, we find that x = 1, x = 3, and x = -5 are roots of the function. Now we know that (x-1), (x-3), and (x+5) are factors of the polynomial.

Step 4: Use polynomial long division or synthetic division to divide f(x) by the factors found in Step 3. In this case, we will divide f(x) by (x-1)(x-3)(x+5).

Step 5: After performing the division, we find that the quotient is (x-1), which means the last root is x = 1.

In conclusion, the zeroes of the function \(f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45\) are x = 1, x = 3, and x = -5.

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The candy store will use 103 grams of sugar in 10 hours.

To find out how many grams of sugar the store will use in 10 hours, we can simply multiply the amount of sugar used in one hour (10.3 grams) by the number of hours (10).

To solve the problem, we use a simple multiplication formula: the amount used per hour (10.3 grams) multiplied by the number of hours (10) to find the total amount of sugar used in 10 hours.

We can interpret this problem using a rate equation: the rate of sugar usage is 10.3 grams/hour, and the time period is 10 hours. Multiplying the rate by the time gives the total amount of sugar used.

So the calculation would be:

10.3 grams/hour x 10 hours = 103 grams

Therefore, the candy store will use 103 grams of sugar in 10 hours.

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The number of units produced increases from 2 to 452 as the number of factory workers increases from 0 to 90, which gives;

Part A:

Part B:

Part C:

Part A:

The correlation is the relationship between variables based on statistical data.

From the given table, the difference between consecutive terms of the x and y-values are constant, therefore as the x-values increases, the corresponding y-value increases.

Change in x-values, ∆x = 10 - 0 = 20 - 10 = 30 - 20 = 10

Change in y-values, ∆y = 52 - 2 = 102 - 52 = 152 - 102 = 50

Therefore;

Part B:

Given that the rate of change of the x-values is constant, and the rate of change of the y-values is a constant, the function relating the x and y-values is a linear function, which can be found as follows;

Slope of the equation, m = ∆y/∆x

Which gives;

m = 50/10 = 5

y - 2 = 5•(x - 0)

The function that best fits the data is therefore;

Part C:

The slope of the function is the coefficient of the variable x in the equation, y = m•x + c

The y-intercept of the function is the value of the constant term, c, in the equation, y = m•x + c

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Answer:

i need more information

Step-by-step explanation:

need more information

Answer:

6h^2 - 19h + 15

Find two numbers that their sum is -19 and their product is 90.

Hence, the numbers will be -9 and -10

6h^2 - 9h - 10h + 15

3h(2h - 3) -5(2h - 3)

(2h - 3)(3h - 5)

Answer:

6x+19

Step-by-step explanation:

The parentheses do nothing here to stop you from adding - since every term in here is only being added, there is no reason to worry about the parentheses.  Simply combine like terms (13 + 6) to achieve 6x + 19.

The appropriate test to use for constructing a confidence interval for the mean household income in the state is the t-test.

The t-test is the appropriate test to use when constructing a confidence interval for the mean household income because the population standard deviation is typically unknown in such cases. The t-test allows for estimating the population standard deviation using the sample standard deviation, making it suitable for situations where the population standard deviation is not known.

To construct a confidence interval, the researcher would typically collect a random sample of household incomes from the state. The sample mean and sample standard deviation are calculated from the data. The t-test uses these sample statistics, along with the desired confidence level and the sample size, to determine the margin of error for the confidence interval.

The margin of error is then added and subtracted from the sample mean to establish the lower and upper bounds of the confidence interval. The t-distribution is used instead of the normal distribution because it accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.

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36 degrees Celsius is equal to 128.8 degrees Fahrenheit.

The conversion from Celsius to Fahrenheit involves using a mathematical formula to convert the temperature from one scale to the other. The formula for converting from Celsius to Fahrenheit is as follows:

Fahrenheit = (Celsius x 9/5) + 32

To perform the conversion, simply substitute the temperature in Celsius into the formula and solve for the temperature in Fahrenheit.

For example, if you want to convert 36 degrees Celsius to Fahrenheit, you would substitute 36 for Celsius in the formula and solve:

Fahrenheit = (36 x 9/5) + 32

Fahrenheit = 96.8 + 32

Fahrenheit = 128.8

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Answer:

A+c

Step-by-step explanation:

because common sense

Answer: The probability of 2 things happening out of 3 is always 1/2

Step-by-step explanation: Because you only want to know the chance of 2 lottery tickets winning, there are only two outcomes 2 tickets winning or not winning

Answer:

You could split 1 into different shapes/parts and later add them up or maybe find the whole area of the figure and subtract the missing pieces/unshaded parts using the given numbers.

Step-by-step explanation:

The answer should be 42 units for both

Answer:

for shape 1 on the left you can do 3x10 +2x5 +1x2 and for shaoe 2 on the right you can do 2x6 +3x5 + 5x3

Check the picture below.

\(\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=\sqrt{a^2 + o^2} \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{9}\\ o=\stackrel{opposite}{8} \end{cases} \\\\\\ c=\sqrt{ 9^2 + 8^2}\implies c=\sqrt{ 81 + 64 } \implies c=\sqrt{ 145 }\implies c\approx 12.04~ft\)

Answer:

Step-by-step explanation:

Given m ∠ Q P S = 4 7 ∘ m∠QPS=47 ∘ m, angle, Q, P, S, equals, 47, degrees m ∠ R P S = 3 x − 3 8 ∘ m∠RPS=3x−38 ∘ m, angle, R, P, S, equals, 3, x, minus, 38, degrees m ∠ Q P R = 7 x − 9 5 ∘ m∠QPR=7x−95 ∘

Sum of angles in a triangle = 180°

47° + 3 x − 3 8 ∘ + 7x - 95° = 180°

10x + 180° = 180°

Answer: 31°

(Attached photo of work)

The probability that K will not be the letter Terry selects is 0.875.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes.

Given that, Jacob mixes the letters J, K, L, J, K, M, N, and P thoroughly.

Here, total number of outcomes = 8

Number of outcomes = 7

Probability of event = 7/8

In decimal

7/8= 0.875

In percentage = 0.875×100

= 87.5%

Therefore, the probability that K will not be the letter Terry selects is 0.875.

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Answer: 4x^6-25

Step-by-step explanation:

1. Distribute

(2x^3+5)(2x^3−5)

2(2^3+5)x^3+5(2x^3-5)

2. Distribute

2(2x^3-5)x^3+5(2x^3-5)

4x^6-10x^3+5(2x^3-5)

3. Distribute

4x^6-10x^3+5(2x^3-5)

4x^6-10x^3+10x^3-25

4. Combine like terms

4x^6-10x^3+10x^3-25

4x^6-25

Answer:

9 roots

Step-by-step explanation:

(x+10)^9 = 0

If you would multiply it out, the highest power is x^9 so the equation has 9 roots

Solving (x+10)^9 =0

Taking the 9th root of each sdie

x+10 = 0

x = -10

The root is -10 with multiplicity of 9

Answer:821

Step-by-step explanation:773+32+7+9=821

The total cost of power generation for a specific load demand can be calculated by optimally allocating the load demand to each unit based on their power output limits. The allocation is done in a way that minimizes the overall cost while meeting the load demand.

In the given power system depicted in Figure 1, there are three units that are always running. Each unit has specific characteristics regarding their minimum power output (Pmin), maximum power output (Pmax), and incremental cost (Ci). Let's discuss the characteristics of each unit and calculate the total cost of power generation for a given load demand.

Unit 1:

Pmin = 200 MW

Pmax = 500 MW

Ci = $50/MWh

Unit 2:

Pmin = 150 MW

Pmax = 400 MW

Ci = $40/MWh

Unit 3:

Pmin = 100 MW

Pmax = 300 MW

Ci = $30/MWh

To calculate the total cost of power generation for a given load demand, we need to determine the optimal power output for each unit. We start by considering the units with the lowest incremental cost first.

Suppose the load demand is D MW. We allocate the load demand to the units as follows:

Step 1: Check if Unit 1 can meet the load demand within its power range. If yes, allocate the load demand to Unit 1 and calculate the cost:

Cost1 = Ci * P1, where P1 is the power output of Unit 1.

Step 2: If there is still remaining load demand, allocate it to Unit 2:

Cost2 = Ci * P2, where P2 is the power output of Unit 2.

Step 3: If there is still remaining load demand, allocate it to Unit 3:

Cost3 = Ci * P3, where P3 is the power output of Unit 3.

Finally, the total cost of power generation, Cost_total, is the sum of Cost1, Cost2, and Cost3:

Cost_total = Cost1 + Cost2 + Cost3

To find the optimal power output for each unit, we consider the load demand and compare it to the minimum and maximum power output limits for each unit. The power allocation is based on meeting the load demand while minimizing the overall cost of power generation.

In summary, given the characteristics of the three units in the power system (Pmin, Pmax, and Ci), the total cost of power generation for a specific load demand can be calculated by optimally allocating the load demand to each unit based on their power output limits. The allocation is done in a way that minimizes the overall cost while meeting the load demand.

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We use ANOVA to test for differences between population means by examining the amount of variability between the samples relative to the amount of variability within the samples. Hence, it is TRUE

The statistical analysis tool known as analysis of variance (ANOVA) divides the observed aggregate variability present in a data set into two categories: systematic variables and random components. The random factors have no statistical impact on the presented data set, whereas the systematic factors do. The ANOVA test is used by analysts to ascertain how independent factors in a regression analysis affect the dependent variable.

ANOVA basically stands for analysis of the variance.

Hence, according to above statements, given fact is true.

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Answer:

3.4561 and 3.4562

Step-by-step explanation:

there are uncountable infinitely many numbers...

Answer:

Answer is 3.456 , 3.456.

Step-by-step explanation:

3.456=3.4560

3.457=3.4570

then numbers between 3.45610 and 3.4570

is - 3.4561 , 3.4562 , ..........., 3.4570

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