3. Each sample of water from a river has a 10% chance of contamination by a particular heavy metal. Find the probability that in 18 independent samples taken from the same river, only two samples were contaminated. [3 marks]​

Answer:

Step-by-step explanation:

To calculate the interest earned by Linda for the first year, we can use the formula:

A = P(1 + r/n)^(nt)

Where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

For the first year, we have:

A = $50,000(1 + 0.06/1)^(1*1) = $53,000

So, the interest earned by Linda for the first year is:

Interest = $53,000 - $50,000 = $3,000

For the second year, we can use the same formula with t = 2:

A = $50,000(1 + 0.06/1)^(1*2) = $56,180

Interest = $56,180 - $53,000 = $3,180

For the third year, we can use the same formula with t = 3:

A = $50,000(1 + 0.06/1)^(1*3) = $59,468.80

Interest = $59,468.80 - $56,180 = $3,288.80

Now, to calculate the interest earned by Bob for each of the first three years, we can use the formula:

Interest = Prt

Where P is the principal amount, r is the annual interest rate, and t is the time in years.

For the first year, we have:

Interest = $50,0000.061 = $3,000

For the second year, we have:

Interest = $50,0000.061 = $3,000

For the third year, we have:

Interest = $50,0000.061 = $3,000

As we can see, Linda earns more interest than Bob for each year, as her interest is compounded annually, while Bob's interest is simple interest. Therefore, the answer is:

Linda earns more.

Answer:

Linda earns $9550.8 interest and bob earns $9000 interest

Step-by-step explanation:

Linda takes compound interest: C.I. = Principal (1 + Rate)Time − Principal

interest= 50,000(1+6/100)³

=59550.8 - 50000

Linda earns $9550.8 interest in 3 years.

bob takes simple interest: S.I = prt/100

interest = 50,000*6*3/100

Bob earns $9000 in 3 years.

thus, Linda earns more interest than bob.

Tom's fortnightly income is $3000.

In mathematics, an average is a measure that represents the central or typical value of a set of numbers. There are several types of averages commonly used, including the mean, median, and mode.

To calculate Tom's fortnightly income, we need to divide his yearly salary by the number of fortnights in a year:

Fortnightly income = Yearly salary / Number of fortnights in a year

Fortnightly income = $78000 / 26 = $3000

Therefore, Tom's fortnightly income is $3000.

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question - Calculate the  Tom's fortnightly income and yearly salary by the number of fortnights in a year .

Answer:

Step-by-step explanation:

Given,

Let length be x and breadth (x - 5).

Perimeter of rectangle is calculated by :

[tex] \: \: \boxed{ \pmb{ \sf{Perimeter_{(rectangle)} = 2(l + b)}}} \\ [/tex]

On substituting the values we get :

[tex]\dashrightarrow \: \: 130 = 2(x + x - 5) \\ [/tex]

[tex]\dashrightarrow \: \: 130 = 2(2x - 5) \\ [/tex]

[tex]\dashrightarrow \: \dfrac{130}{2} = (2x - 5) \\ [/tex]

[tex]\dashrightarrow \: \: 65 = 2x - 5 \\ [/tex]

[tex]\dashrightarrow \: \: 65 + 5 = 2x \\ [/tex]

[tex]\dashrightarrow \: \: 70 = 2x \\ [/tex]

[tex]\dashrightarrow \: \: \frac{70}{2} = x \\ [/tex]

[tex]\dashrightarrow \: \: 35 = x \\ [/tex]

Hence,

Using binomial theorem, the generating function is G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256 while the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.

To find the generating function of (x+4)^4, we expand it using the binomial theorem:

[tex](x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)[/tex]

where C(n,k) denotes the binomial coefficient "n choose k".

Simplifying the terms, we get:

[tex](x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]

Therefore, the generating function of (x+4)^4 is:

[tex]G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]

The associated sequence can be read off by finding the coefficients of each power of x:

To find the generating function of (x+4)^4, we expand it using the binomial theorem:

(x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)

where C(n,k) denotes the binomial coefficient "n choose k".

Simplifying the terms, we get:

(x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256

Therefore, the generating function of (x+4)^4 is:

G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256

The associated sequence can be read off by finding the coefficients of each power of x:

The coefficient of x^k is the k-th term of the sequence.

In this case, the sequence is given by the coefficients of G(x):

a₀ = 256

a₁ = 256

a₂ = 96

a₃ = 16

a₄ = 1

Therefore, the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.

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Answer:

A. To find the derivative of h(x), we can use the chain rule:

h(x) = a(-2x + 1)^5 - b

h'(x) = a * 5(-2x + 1)^4 * (-2) = -10a(-2x + 1)^4

To find the second derivative, we can again use the chain rule:

h''(x) = -10a * 4(-2x + 1)^3 * (-2) = 80a(-2x + 1)^3

B. To show that h is monotonic, we need to show that h'(x) is either always positive or always negative. Since h'(x) is a multiple of (-2x + 1)^4, which is always non-negative, h'(x) is always either positive or negative depending on the sign of a. If a > 0, then h'(x) is always negative, which means that h(x) is decreasing. If a < 0, then h'(x) is always positive, which means that h(x) is increasing.

C. To find the critical points, we need to find where h'(x) = 0:

h'(x) = -10a(-2x + 1)^4 = 0

-2x + 1 = 0

x = 1/2

Thus, the critical point is at x = 1/2. This value is independent of a and b, as neither a nor b appear in the calculation of the critical point.

An equivalent cοmpοund inequality is -9 < x < -3.

An inequality with an absοlute value algebraic expressiοn and variables is knοwn as an absοlute value inequality. An expressiοn using absοlute functiοns and inequality signs is knοwn as an absοlute value inequality.

Here, we have

Given: |x+6|<3 is an absοlute value inequality.

Apply absοlute rule

if |u|<a, a>0 then -a<u<a

-3 < x+6 <3

x+6 > -3 and x+6 <3

x> -9 and x< -3

-9 < x < -3

Hence, an equivalent cοmpοund inequality is -9 < x < -3.

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The Sunday game brought in $2232.50, while the Thursday game brought in $1832.00.

A company's income, costs, and profit are compiled in a profit and loss (P&L) statement, a financial report. It provides information to investors and other interested parties about a company's operations and financial viability.

The issue informs us that the combined revenue from the two games was $4064.50.

S + (S - 400.50) = 4064.50

Simplifying the left side, we get:

2S - 400.50 = 4064.50

Adding 400.50 to both sides, we get:

2S = 4465

Dividing both sides by 2, we get:

S = 2232.50

So the Sunday game generated $2232.50, and the Thursday game generated $2232.50 - $400.50 = $1832.00.

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Answer: 6y + 6

Step-by-step explanation:

To simplify the expression 8(3/4y -2) + 6(-1/2+4) + 1, we can follow the order of operations (PEMDAS):

First, we simplify the expression within parentheses, working from the inside out:

6(-1/2+4) = 6(7/2) = 21

Next, we distribute the coefficient of 8 to the terms within the first set of parentheses:

8(3/4y -2) = 6y - 16

Finally, we combine the simplified terms:

8(3/4y -2) + 6(-1/2+4) + 1 = 6y - 16 + 21 + 1 = 6y + 6

Therefore, the expression 8(3/4y -2) + 6(-1/2+4) + 1 is equivalent to 6y + 6.

Answer:

D I believe

Step-by-step explanation:

The probability of obtaining a reading less than 0.35° C is approximately 35%.

Accοrding tο the prοbability fοrmula, the likelihοοd οf an event οccurring is equal tο the ratiο οf the number οf favοurable οutcοmes tο the tοtal number οf οutcοmes. Prοbability οf an event οccurring P(E) = The number οf favοurable οutcοmes divided by the tοtal number οf οutcοmes.

The readings at freezing οn a set οf thermοmeters are nοrmally distributed, with a mean (x) οf 0°C and a standard deviatiοn (μ) οf 1.00°C. We want tο knοw hοw likely it is that we will get a reading that is less than 0.35°C.

To solve this problem, we must use the z-score formula to standardise the value:

[tex]$Z = \frac{x - \mu}{\sigma}[/tex]

Z = standard score

x = observed value

[tex]\mu[/tex] = mean of the sample

[tex]\sigma[/tex] = standard deviation of the sample

Here

x = 0.35° C

[tex]\mu[/tex] = 0° C

[tex]\sigma[/tex] = 1.00°C

Using the values on the formula:

[tex]$Z = \frac{0.35 - 0}{1}[/tex]

Z = 0.35

The probability of obtaining a reading less than 0.35° C is approximately 35%.

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Step-by-step explanation:

(-1, 1) is a solution.

because

-(-1) + 6×1 = 7

1 + 6 = 7

7 = 7

correct.

every ordered pair of x and y values that make the equation true is a solution.

(5, 2) would be another solution. and so on.

Using the concept of parabola, the quadratic inequality represented by the graph can be written as:

y = x² -2x +2.

An equation of a curve that has a point on it that is equally spaced from a fixed point and a fixed line is referred to as a parabola.

The parabola's fixed point is referred to as the focus, and its fixed line is referred to as the directrix.

The general equation for a parabola is given as:

y = a(x-h) ² + k

Now here we have:

(x,y) = (2,5)

(h,k) = (1,1)

Putting these values in the equation,

5 = a (2-1) ² + 1

a = 5-1

=4

Substituting the values:

y = (x-1) + 1

y = x² -2x +2

Therefore, the quadratic inequality can be written as: y = x² -2x +2.

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Step-by-step explanation:

B) 20:23

C) 6,0375

hope this helps

Answer:

55 degrees angles on a rights angle triangle. 1 and 3 they are equal cause they are vertical opp angles 55 degrees

Step-by-step explanation:

4)

based on similar triangles and the common ratio for all pairs of corresponding sides we know

LE/LM = LD/LK = DE/EM

because E and D are the midpoints of the longer sides, all of these ratios are 1/2.

1/2 = DE/8

8/2 = 4 = DE

5)

same principle as for 4)

BQ/BA = BR/BC = QR/AC

again, Q and R are the midpoints, so all these ratios are 1/2.

1/2 = QR/10

QR = 10/2 = 5

In response to the stated question, we may state that As a result, the 66th derivative of f(x) = 4 sin(x) is 4 sin(x) (x).

In mathematics, the derivative of a function with real variables measures how sensitively the function's value varies in reaction to changes in its parameters. Derivatives are the fundamental tools of calculus. Differentiation (the rate of change of a function with respect to a variable in mathematics) (in mathematics, the rate of change of a function with respect to a variable). The use of derivatives is essential in the solution of calculus and differential equation problems. The definition of "derivative" or "taking a derivative" in calculus is finding the "slope" of a certain function. Because it is frequently the slope of a straight line, it should be enclosed in quotation marks. Derivatives are rate of change metrics that apply to almost any function.

Using the chain rule and the derivative of the sine function repeatedly yields the 66th derivative of the function [tex]f(x) = 4 sin (x).[/tex]

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x), and this pattern repeats itself every two derivatives.

As a result, the first derivative of f(x) is:

[tex]f'(x) = 4 cos (x)[/tex]

The second derivative is as follows:

[tex]f"(x) = -4 sin (x)[/tex]

The third derivative is as follows:

[tex]f"'(x) = -4 cos (x)[/tex]

The fourth derivative is as follows:

[tex]f""(x) = 4 sin (x)[/tex]

And so forth.

[tex]f^{(66)(x)} = 4 sin (x)[/tex]

Because the pattern repeats every four derivatives, the 66th derivative is the same as the second, sixth, tenth, fourteenth, and so on.

As a result, the 66th derivative of f(x) = 4 sin(x) is 4 sin(x) (x).

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The scale of the map = 682.5.

We kept a safe distance and followed them. She perceives a separation between her and her brother that wasn't there before. Although they were previously close friends, there was now a great deal of gap between them.

We must calculate the ratio of the distance shown on the map to the real distance between the cities in order to ascertain the scale of the map.

We are aware that there are 210 miles separating the two cities. Let x represent the precise location of this distance on the map. From that, we may establish the ratio:

Actual distance / Map Distance = 210 / x

The distance on the map is indicated as 3 1/4 inches, which is also known as 13/4 inches. When we enter this into the percentage, we obtain:

Actual distance divided by (13/4) = 210 / x

We can cross-multiply and simplify to find x's value:

Actual distance: 682.5 = x * 210 x = 3.25 when 210 * (13/4) Equals x.

Consequently, 3.25 inches on the map represent the actual distance between the cities. We can write: To determine the map's scale:

Actual distance divided by 1 inch on the chart equals 210 miles.

When we replace the values we discovered earlier, we obtain:

1 / 210 = 3.25 / scale

If we solve for the scale, we obtain:

scale = 682.5.

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To sketch the phase plane and trajectories of both populations in the competing species model, plot the population of one species on the x-axis and the population of the other species on the y-axis. Then, plot the isoclines and use them to determine the direction and stability of the population trajectories.

The competing species model is a system of two differential equations that describe the population dynamics of two species competing for the same resources. To sketch the phase plane and trajectories, plot the population of one species on the x-axis and the population of the other species on the y-axis. Then, plot the isoclines, which are curves that represent the values of one species' population at which the other species' population does not change.

The isoclines are found by setting each differential equation to zero and solving for one population in terms of the other. For example, the isocline for species 1 is found by setting dN1/dt = 0 and solving for N2. The resulting equation gives the values of N2 at which the population of species 1 does not change. Plotting these curves on the phase plane divides it into regions where the population of each species increases or decreases.

The direction and stability of the population trajectories can be determined by analyzing the slope of the vector field, which represents the rate of change of the population at each point in the phase plane. Trajectories move in the direction of the vector field, and their stability depends on the curvature of the isoclines. If the isoclines intersect at a single point, it is a stable equilibrium where both populations coexist. If they intersect at multiple points, the stable equilibrium depends on the initial conditions of the populations. If they do not intersect, one species will eventually drive the other to extinction.

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--The question is incomplete, answering to the question below--

"Consider the competing species model, how to sketch the phase plane and the trajectories of both population"

Answer:

3x(x - 4y)

Step-by-step explanation:

3x² - 12xy ← factor out 3x from each term

= 3x(x- 4y)

Answer:

£22

Step-by-step explanation:

50% of 88=88/100 ×50=44

44÷2=25%=22

75% of £88 is deducted, so that 88-66=£22

Answer:

Step-by-step explanation:

To find the number of students who studied in the library or cafeteria (including both), we need to add the number of students who studied in the library and the number of students who studied in the cafeteria, but we need to subtract the number of students who studied in both the library and cafeteria to avoid counting them twice.

So, the number of students who studied in library or cafeteria is:

170 + 135 - 76 = 229

Therefore, 229 students studied in the library or cafeteria (including both).

The difference in length between the two pieces of scrap wood is 7/8 inches.

To get the difference we just need to take the difference between the two lenghs.

Remember that we only have pieces of scraph wood if we have an "x" over the correspondent value in the line diagram.

By looking at it we can see that the longest pice measures 5 inches, while the shortest one (there are two of these) measure (4 + 1/8) inches.

The difference is:

5 - (4 + 1/8) = 7/8

The longest piece is 7/8 inches longer.

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Answer:

The given function is an exponential growth function, not an exponential decay function because as the exponent x increases, the value of y also increases instead of decreasing.

To describe its end behavior using limits, we need to find the limit of the function as x approaches infinity and as x approaches negative infinity.

As x approaches infinity, the exponent -x approaches negative infinity, and the base (1/6) is raised to increasingly larger negative powers, causing the function to approach zero. So, the limit as x approaches infinity is 0.

As x approaches negative infinity, the exponent -x approaches infinity, and the base (1/6) is raised to increasingly larger positive powers, causing the function to approach infinity. So, the limit as x approaches negative infinity is infinity.

Therefore, the end behavior of the function is that it approaches zero as x approaches infinity and approaches infinity as x approaches negative infinity.

Answer:Approximately

(0.58975,0.34781)

Step-by-step explanation:

If (x,y) is a point on the parabola, then the distance between (x,y) and (1,0) is:

√(x−1)2+(y−0)2=√x4+x2−2x+1

To minimize this, we want to minimize

f(x)=x4+x2−2x+1

The minimum will occur at a zero of:

f'(x)=4x3+2x−2=2(2x3+x−1)

graph{2x^3+x-1 [-10, 10, -5, 5]}

Using Cardano's method, find

x=3√14+√8736+3√14−√8736≅0.58975

y=x2≅0.34781

(a) The function x as a function of t is t = 250ln(499x/998)

(b) Half the population will have heard the rumor approximately 109.86 units of time after it was started.

(a) To solve the differential equation dt/dx = kx(N−x), we can separate the variables and integrate

dt/dx = kx(N−x)

dt/(N-x) = kx dx

Integrating both sides, we get

t = -1/k × ln(N-x) - 1/k × ln(x) + C

where C is the constant of integration.

To find C, we can use the initial condition that two people start the rumor at t=0, so x=2:

0 = -1/k * ln(N-2) - 1/k * ln(2) + C

C = 1/k * ln(N-2) + 1/k * ln(2)

Substituting C back into the equation, we get:

t = -1/k * ln(N-x) - 1/k * ln(x) + 1/k * ln(N-2) + 1/k * ln(2)

Simplifying, we get

t = 1/k * [ln((N-2)x/(2(N-x)))]

Substituting k=1/250 and N=1000, we get:

t = 250ln(499x/998)

(b) We want to find the time t when half the population has heard the rumor, so x = N/2 = 500. Substituting this into the equation we obtained in part (a), we get

t = 250ln(499(500)/998) = 250ln(249/499)

t ≈ 109.86

Therefore, half the population will have heard the rumor approximately 109.86 units of time after it was started.

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Graph D of the given function has the smallest value for b.

As per name signifies, exponents are used in exponential functions. But take note that an exponential function does not have a constant as its base and a variable as its exponent. One of the following forms can be used for an exponential function.

f (x) = aˣ

y=f(x) >0

f(x)=abˣ , where a>0

So, f(x)=abˣ

When, b<1 f(x) decreases

When, b>1 f(x) increases and the larger the b the steeper the graph

So, graph of D is increasing and is steepest

So, graph D has the smallest value for b.

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Answer:

Step-by-step explanation:

To find the current price index number using the 1975 price of 56.7 cents as the reference value, we can use the formula:

Price Index = (Current Price / Base Price) x 100

Where "Current Price" is the current cost of gasoline, and "Base Price" is the 1975 price of 56.7 cents.

Substituting the values given in the problem, we get:

Price Index = ($2.93 / $0.567) x 100

Price Index = 516.899

Therefore, the current price index number, using the 1975 price of 56.7 cents as the reference value, is 516.899.

The standard normal areas are given as follows:

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Considering the second bullet point, the areas are given as follows:

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