b) During the first market day, Fatuma bought 30 oranges and 12 mangoes and paid Ksh. 936 for all the fruits. In the second market day, the price of an orange increased by 20% while that of a mango reduced in the ratio 3:4. Fatuma bought 15 oranges and 20 mangoes and paid Ksh. 780 for all the fruits. Given that the cost of an orange and that of a mango during the first market day was Ksh. x and Ksh. y respectively: (i) Write down simultaneous equations to represent the information above. (2 marks) (ii) Use matrix in (a) above to find the cost of an orange and that of a mango in the first market day. (4 marks) (iii) Fatuma sold all the fruits bought on the second market day at a profit of 10% per orange and 15% per mango. Calculate the total amount of money realized for the sales. (2 marks)​

Mr. Bilo received 8 P50-bills, 5 P100-bills, and 5 P20-bills.

The number of P50, P100, and P20 bills Mr. Pilo got is calculated as follows;

Amount from P50-bills: 2/5 x P1,000 = P400

Amount from P100-bills: 1/2 x P1,000 = P500

The total amount he received from the P50-bills and P100-bills = P400 + P500

The total amount he received from the P50-bills and P100-bills = P900. The amount left to be changed into P20-bills = P1,000 - P900

The amount left to be changed into P20-bills = P100.

The number of bills will then be:

Number of P50-bills: P400 ÷ P50 = 8

Number of P100-bills: P500 ÷ P100 = 5

Number of P20-bills: P100 ÷ P20 = 5

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Complete question:

Mr Bilo has a P1,000-bill and asked someone to exchange in to 2/5 P50-bill and 1/2 P100-bill and the rest into 20-bill. How many P50,P100 and P20 bills did he get.

Hunter has 32 different possible outcomes if he rolls a pyramid-shaped die with four sides, spins a spinner with four equal-sized sections, and flips a coin.

There are different methods to approach this problem, but one possible way is to use the multiplication principle of counting, which states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks in sequence.

In this case, Hunter has three tasks: rolling the die, spinning the spinner, and flipping the coin.

For the first task, rolling the die, there are four possible outcomes.

For the second task, spinning the spinner, there are four possible outcomes as well.

For the third task, flipping the coin, there are two possible outcomes.

Using the multiplication principle, the total number of possible outcomes is:

4 x 4 x 2 = 32

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Rate of change in the temperature of lasagna given by function T(t) = 70 + (200 - 70) × [tex]e^{(-0.0001t)}[/tex] exactly 2 hours after taking it out of oven is -0.013 °F/hour.

Surrounding room temperature is equal to 70°F

Temperature at which lasagna comes out of the oven = 200°F

The temperature T (in °F) of the lasagna at time t (in hours) after taking it out of the oven is equal to,

T(t) = 70 + (200 - 70) × [tex]e^{(-0.0001t)}[/tex]

To find the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven,

Find the derivative of the temperature function with respect to time t.

T'(t) = -0.013[tex]e^{(-0.0001t)}[/tex]

Substituting t = 2 into this expression gives:

T'(2) = -0.013[tex]e^{(-0.0001\times 2)}[/tex]

= -0.013[tex]e^{-0.0002}[/tex]

= -0.013 × 0.99980

= -0.0129974

= -0.013

Therefore, the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven is approximately -0.013 °F/hour.

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The given question is incomplete, I answer the question in general according to my knowledge:

A tray of lasagna comes out of the oven at 200°F and is placed on a table where the surrounding room temperature is 70°F. The temperature T (in °F) of the lasagna is given by the function T(t) = 70 + (200 - 70) × [tex]e^{(-0.0001t)}[/tex] where t is time(in hours) after taking the lasagna out of the oven. What is the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven?

Answer:

Step-by-step explanation:

1 - divide

2 - measur

3 - add

answer = 44.78

The volume of the solid obtained by rotating the region bounded about the x-axis is 3π/4 cubic units.

To find the volume of the solid obtained by rotating the region bounded by y = 4x^2, y = 1, and y = 0 about the x-axis, we can use the method of cylindrical shells.

First, we need to find the limits of integration. The region is bounded by y = 4x^2 and y = 1, so we can set up the integral as follows:

V = ∫[0,1] 2πx(1-4x^2)dx

Next, we can simplify the integrand:

V = ∫[0,1] 2πx dx - ∫[0,1] 8πx^3 dx

V = π - 2π/4

V = 3π/4

Therefore, the volume of the solid obtained by rotating the region bounded by y = 4x^2, y = 1, and y = 0 about the x-axis is 3π/4 cubic units.

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The given pattern involves adding 3 to the previous term, so the correct function would be: an = an-1 + 3.

To determine the function that represents a given sequence pattern, we must first examine the relationship between the terms in the sequence.

In this case, the pattern involves adding 3 to the previous term to obtain the next term.

Therefore, the correct function would be of the form an = an-1 + 3, where "an" represents the current term in the sequence and "an-1" represents the previous term. This recursive function defines the relationship between each term in the sequence and can be used to find any term in the sequence, given the previous term.

It is important to identify the correct function that represents a given sequence pattern, as it can be used to find any term in the sequence, as well as to predict future terms. This can be particularly useful in fields such as finance and economics, where analyzing patterns in data is a critical component of decision-making.

The given pattern involves adding 3 to the previous term, so the correct function would be: an = an-1 + 3

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The set of all points in quadrant I not including the axis's can be represented by the following system of inequalities:

x > 0

y > 0

Inequalities are useful in modeling situations where there are constraints or limitations. For  illustration, in real- life  scripts, there may be limited  coffers or capacity, or certain variables must fall within a specific range. Systems of inequalities are  frequently used to represent these constraints or limitations graphically.   One common  operation of systems of inequalities is in optimization problems, where the  thing is to maximize or minimize a particular function subject to certain constraints.

In these situations, the  doable region, or the set of all points that satisfy the constraints, is  frequently represented as a shadowed region on a graph. The optimal  result is  also  set up by  relating the point( s) within this region that maximize or minimize the function.

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It would take approximately 95 minutes for the wick to have 74 mm remaining.

1) Finding the slope (mm per minute):
The wick was initially 150 mm in length and reduced to 110 mm after 50 minutes. To find the slope, we use the formula:

Slope = (change in length) / (change in time)

Slope = (110 mm - 150 mm) / (50 minutes - 0 minutes)
Slope = (-40 mm) / (50 minutes)
Slope = -0.8 mm/minute

2) Constructing the linear equation:
We now have the slope (-0.8) and the initial length (150 mm) to create a linear equation:

Length (L) = initial length + slope × time (t)
L = 150 - 0.8t

3) Predicting the time to have 74 mm remaining:
To find the time, plug in 74 mm for the length (L) in the equation and solve for t:

74 = 150 - 0.8t
76 = 0.8t
t = 95 minutes

So, it would take approximately 95 minutes for the wick to have 74 mm remaining.

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2/5 of the packages weighed more than 1 2/5 pounds.

To find the fraction of packages that weighed more than 1 2/5 pounds, we need to count the number of packages that weighed more than 1 2/5 pounds and divide it by the total number of packages.

There are a total of 10 packages. Out of these, we can see that 4 packages weighed more than 1 2/5 pounds: 1 1/5, 1 3/5, 1 3/5, and 1 4/5.

Therefore, the fraction of packages that weighed more than 1 2/5 pounds is:

4/10

This can be simplified to:

2/5

Hence, only 2/5 of the package weighed more.

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Answer:

76

Step-by-step explanation:

If the mean of 28 numbers is 18 then the sum of those numbers=28×18=504

if one number is added then 29×20=580

the new number therefore=580-504=76

The probability that on a particular Monday, they each suggest a different activity is 0.11.

The probability that in a period of four consecutive Mondays, they all suggest the same activity on exactly two of the four Mondays is 0.0504.

1. Probability that on a particular Monday, they each suggest a different activity:

The probability is calculated using the formula below:

Probability = P(SMC) + P(MCS) + P(CSM)

Probability = (0.4 x 0.55 x 0.5) + (0.3 x 0.2 x 0.3) + (0.3 x 0.25 x 0.2)

Probability = 0.11

2. The probability that in a period of four consecutive Mondays, they all suggest the same activity on exactly two of the four Mondays is determined using the binomial distribution.

Let success be suggesting the same activity on exactly two of the four Mondays.

The probability of success on any Monday is:

P(success) = P(SSNN) + P(NSSN) + P(NNSS)

P(success) = 3 x (0.4 x 0.4 x 0.6 x 0.6)

P(success) = 0.3456

The probability of failure is:

P(failure) = 1 - P(success)

P(failure)= 1 - 0.3456

P(failure) = 0.6544

Choose exactly two Mondays out of four is ⁴C₂

The probability of exactly two successes = ⁴C₂ * P(success)² * P(failure)²

P(exactly 2 successes) = 6 x (0.3456)² x (0.6544)²

P(exactly 2 successes) = 0.0504 or 5.04%

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Answer:

11.86 units

Step-by-step explanation:

Volume = length x width x height

We know that the volume is 210 and the width is 5 3/5 (or 5.6) and the height is 3 1/3 (or 3.33).

Substituting these values into the formula, we get:

210 = length x 5.6 x 3.33

To solve for the length, we can divide both sides of the equation by (5.6 x 3.33):

length = 210 / (5.6 x 3.33)

Simplifying this expression, we get:

length ≈ 11.86

Therefore, the length of the large box is approximately 11.86 units.

To find out how much Peter eats in one week (which is 7 days), we need to multiply this expression by 7.

Peter eats 3 carrot sticks and 1 cup of peanut butter before lacrosse practice every day, so in one day he eats:

3 + p

To find out how much he eats in one week (which is 7 days), we need to multiply this expression by 7:

7(3 + p)

Distributing the 7, we get:

21 + 7p

So the equivalent expressions that represent how much Peter eats before practice in one week are:

3 + 4p + 3p

4(3 + p)

21 + 7p

7(3p + 1)

So the correct answers are:

4(3 + p)

21 + 7p

7(3p + 1)

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The radius of convergence is 1/3. To find the radius of convergence for the power series using the ratio test, we need to analyze the general term of the series. The given series is:

Σ(an * x^n)

where an is the coefficient of the term with x raised to the power n. The coefficients in the given series are:

a1 = 3
a2 = 36
a3 = 243
a4 = 1296
a5 = 6075
...

Notice that each coefficient is a multiple of 3^n. Thus, we can write the general term as:

an = 3^n

Now, we apply the ratio test. The ratio test states that the series converges if the limit as n approaches infinity of the absolute value of the ratio of consecutive terms is less than 1:

lim (n → ∞) |(a(n+1) * x^(n+1)) / (an * x^n)|

= lim (n → ∞) |(3^(n+1) * x^(n+1)) / (3^n * x^n)|

To simplify, divide 3^(n+1) by 3^n:

= lim (n → ∞) |(3 * x)|

The series converges when |3 * x| < 1. To find the radius of convergence, solve for |x|:

|x| < 1/3

The radius of convergence is 1/3.

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The area of the shaded region of the circle is 89.75 mi².

The area of a sector of a circle, you can use the formula:

A = (θ/360) × π × r²

Where A is the area of the sector, θ is the central angle of the sector, r is the radius of the circle, and π is a constant approximately equal to 3.14.

From the diagram, angle of the unshaded sector equals 150 degree.

Angle of the shaded region = 360 - 150 = 210 degree

Radius r = 7 miles.

We can substitute these values into the formula and solve for the area A.

A = (θ/360) × π × r²

A = ( 210/360 ) × 3.14 × 7²

A = ( 210/360 ) × 3.14 × 49

A = 89.75 mi²

Therefore, the area of the sector is approximately 89.75 mi².

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The volume of the solid will be 32/7 √3.

Since base of a solid whose cross-sections is perpendicular to the w-axis are equilateral triangles

Now base of triangle. is f(x) = x³ and the Area of Equilateral Triangle is ✓3/4 base²

The volume of the solid will be:

= ✓3/4 (x^7/7)²

= ✓3/4 (128/7)

= 32/7 √3

Therefore, the volume of the solid will be 32/7 √3.

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The perimeter of the track is of 377 m.

The circumference of a circle of radius r is given by the equation presented as follows:

C = 2πr.

Considering the rectangle with area 5500 m² and base 110 m, the height h, representing the diameter d of the circumference, is obtained as follows:

110d = 5500

d = 550/11

d = 50 m.

The radius is half the diameter, hence it is given as follows:

r = 25 m.

The perimeter is given as follows:

Hence it is given as follows:

P = 2 x 110 + 2 x 3.14 x 25

P = 377 m.

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The only possible value of m that would make the ranges of f(x) and g(x) the same is any positive real number.

The range of a function is the set of all possible output values. In this case, we are given that the ranges of two functions, f(x) and g(x), are the same.

The function f(x) = √(mx) has a domain of x ≥ 0, since the square root of a negative number is not a real number. The function g(x) = m√x has a domain of x ≥ 0 for the same reason.

To find the range of these functions, we need to consider the possible values of the input x. For f(x), as x increases, the output √(mx) also increases, and as x approaches infinity, so does the output. For g(x), as x increases, the output m√x also increases, and as x approaches infinity, so does the output.

Therefore, if the ranges of f(x) and g(x) are the same, this means that they both have the same maximum and minimum values, and these values are achieved at the same inputs.

In particular, if we consider the minimum value of the range, this is achieved when x = 0, since both functions are defined only for non-negative inputs. At x = 0, we have f(0) = g(0) = 0, so the minimum value of the range is 0.

To find the maximum value of the range, we need to consider the behavior of the functions as x approaches infinity. As noted above, both functions increase without bound as x increases, so the maximum value of the range is infinity.

Therefore, the only possible value of m that would make the ranges of f(x) and g(x) the same is any positive real number.

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To estimate a population mean with 68.26% confidence that the sample mean will not differ from the population mean by more than 4 units, a sample size of 7 is needed. So, the correct answer is B).

The formula to calculate the sample size needed to estimate the population mean with a specified margin of error, assuming the population standard deviation is known, is

n = ((z-score * σ) / E)²

where

n = sample size

z-score = the z-score corresponding to the desired confidence level (in this case, the 68.26% confidence level corresponds to a z-score of 1)

σ = population standard deviation

E = the desired margin of error

Substituting the given values, we get

n = ((1 * √(99)) / 4)²

n = 6.1875

Since we need to have a whole number for the sample size, we must round up to the nearest integer. Therefore, the necessary sample size is 7.

So, the answer is B) 7.

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The probability of flipping a coin and getting tails at least once in four flips is 15/16 or approximately 0.94. (option d).

To determine the probability of flipping a coin and getting tails at least once in four flips, we can use a probability table. The table shows all the possible outcomes of flipping a coin four times.

Flip 1 Flip 2 Flip 3 Flip 4

Outcome 1 H H H H

Outcome 2 H H H T

Outcome 3 H H T H

Outcome 4 H H T T

Outcome 5 H T H H

Outcome 6 H T H T

Outcome 7 H T T H

Outcome 8 H T T T

Outcome 9 T H H H

Outcome 10 T H H T

Outcome 11 T H T H

Outcome 12 T H T T

Outcome 13 T T H H

Outcome 14 T T H T

Outcome 15 T T T H

Outcome 16 T T T T

In the table, H represents heads, and T represents tails. There are 16 possible outcomes when flipping a coin four times. We can see that getting tails at least once is possible in 15 of these outcomes: Outcome 2, Outcome 3, Outcome 4, Outcome 6, Outcome 7, Outcome 8, Outcome 10, Outcome 11, Outcome 12, Outcome 14, Outcome 15, and Outcome 16.

Therefore, the probability of flipping a coin and getting tails at least once in four flips is the number of outcomes where tails appear at least once divided by the total number of outcomes, which is 15/16 or approximately 0.94. (option d).

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The horizontal distance between the solutions is -12/5.

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function is y = [tex]-5x^2+12x[/tex].

Now take y=0 then,

=>  [tex]-5x^2+12x[/tex] = 0

=> x(-5x+12) = 0

=> x = 0 and -5x = 12

=> [tex]x_1[/tex] = 0 and [tex]x_2[/tex] = -12/5

Then horizontal distance = [tex]x_2-x_1[/tex]

=> [tex]\frac{-12}{5}-0[/tex] = -12/5

Hence the horizontal distance between the solutions is -12/5.

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Answer:

Step-by-step explanation:

hope this helps you

Ion has a total of $4,378.05 in his savings account at the end of 3 years.

To solve this problem, we need to use the formula for compound interest:

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

Where:

A = the total amount of money in the account at the end of the time period

P = the principal amount (initial deposit)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period (in years)

In this case, we have:

P = $120

r = 0.029 (2.9% expressed as a decimal)

n = 12 (compounded monthly)

t = 3 (years)

We also need to calculate the total number of deposits made during the 3-year period. Since a deposit of $120 is made at the end of every month, and there are 12 months in a year, the total number of deposits made is:

12 deposits/year × 3 years = 36 deposits

Now we can plug in the values into the formula:

A = [tex]120(1 + 0.029/12)^(12 × 3) + 120[(1 + 0.029/12)^(12 × 3) - 1]/(0.029/12)[/tex]

A = $4,378.05

Therefore, Ion has a total of $4,378.05 in his savings account at the end of 3 years.

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Answer:

The answer is the fourth option - 0.63

Step-by-step explanation:

The function is neither even nor odd.

To find f(-x), we substitute -x for x in the function f(x):

f(-x) = (-x)^3 - 7/(-x) = -x^3 - 7/x

To find -f(x), we multiply the function f(x) by -1:

-f(x) = -1 * (x^3 - 7/x) = -x^3 + 7/x

To determine if the function is even, odd or neither, we compare f(-x) and -f(x).

If f(-x) = f(x), the function is even.

If f(-x) = -f(x), the function is odd.

If neither of these is true, the function is neither even nor odd.

Comparing f(-x) and -f(x), we have:

f(-x) = -x^3 - 7/x

-f(x) = -x^3 + 7/x

Since f(-x) and -f(x) are not equal, and f(-x) is not the negative of -f(x), the function is neither even nor odd.

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This is the derivative of f(x) with respect to x:

f'(x) = (-2e^(-x) + 1/x)·(9x^2.5) + (2e^(-x) + ln(x))·(22.5x^1.5)

To find the derivative of f(x) = g(x) · h(x), we'll use the product rule, which states that (u·v)' = u'·v + u·v'. Let u = g(x) and v = h(x).

u = g(x) = 2e^(-x) + ln(x)
v = h(x) = 9x^2.5

Now, find the derivatives of u and v:

u' = g'(x) = -2e^(-x) + 1/x
v' = h'(x) = 22.5x^1.5

Now apply the product rule:

f'(x) = u'·v + u·v'
f'(x) = (-2e^(-x) + 1/x)·(9x^2.5) + (2e^(-x) + ln(x))·(22.5x^1.5)

This is the derivative of f(x) with respect to x.

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The Alverados spend $1,200 more on taxes than on clothing.

To determine how much more the Alverados spend on taxes than on clothing, we need to know how much they spend on each.

If we assume that the Alverados spend 25% of their income on taxes, that would be:

0.25 x $6,000 = $1,500

If we assume that the Alverados spend 5% of their income on clothing, that would be:

0.05 x $6,000 = $300

To find how much more they spend on taxes than on clothing, we can subtract the amount spent on clothing from the amount spent on taxes:

$1,500 - $300 = $1,200

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