please help if you can

The probability that Peggy can exchange for a gift coupon is  36.3%.

To find the probability that Peggy can exchange for a gift coupon, we need to calculate the probability that she obtains at least 20 reward points from the 4 packets she buys.

Let X be the total number of points Peggy obtains from the 4 packets. We can model X as a binomial random variable with n = 4 and p = 0.75, since 75% of the packets contain 3-point cards.

The probability mass function of X is:

P(X = k) = (4 choose a) * 0.75ᵃ* 0.25⁴⁻ᵃ, for a = 0, 1, 2, 3, 4

To find the probability that Peggy can exchange for a gift coupon, we need to calculate P(X >= 20). However, since the number of points in each packet is discrete and limited to 3 or 7, it is not possible to obtain exactly 20 points. Therefore, we need to find P(X >= 21).

P(X >= 21) = P(X = 21) + P(X = 22) + P(X = 23) + P(X = 24) + P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28)

Using the binomial probability mass function, we get:

P(X >= 21) = (4 choose 3) * 0.75³ * 0.25 + (4 choose 2) * 0.75² * 0.25² + (4 choose 1) * 0.75 * 0.25³ + 0.25⁴

P(X >= 21) = 0.36328125

Therefore, the probability that Peggy can exchange for a gift coupon is approximately 0.363, or 36.3%.

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The simulatiοn dοes't behave in way that is similar tο actual dice because οf variatiοn οf prοbability.

Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.

a)The pοssible cοmbinatiοns frοm the twο dices tο get a tοtal οf seven are

=> (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)

i.e. 6 cases and the tοtal nο. οf cοmbinatiοns

=>6 * 6 = 36.

Hence the nο. οf cases favοurable fοr us οut οf all the cοmbinatiοns is  6.

Hence the prοbability is  6/36 = 1/6.

b)To get a 7 out from a set of 2,12 inclusive is 1/11.

Because it is a choice of 1 no. from the 11 numbers available.

c). NO.

This simulatiοn dοesn't behave like the actual dice because in the secοnd case we are chοοsing οne number frοm the 11 available οptiοns whereas in the first case it is the prοbability οf chοοsing the 6 right cοmbinatiοns frοm the 36 cοmbinatiοns available.

i.e . Chοοsing 6 frοm 36 . hence the prοbability is 1/6.

whereas the chοice οf 1 frοm 11 . i.e prοbability is 1/11.

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Using the formula of margin of error for the confidence interval, the minimum sample size is 120

We can use the formula for the margin of error for a confidence interval:

margin of error = z* (standard deviation/√n)

where z* is the z-score for the desired confidence level.

For a 95% confidence interval, z* = 1.96. We want the margin of error to be 5, and we are given that the standard deviation of the population is 28. So we can solve for n:

5 = 1.96 * (28 / √n)

√n = 1.96 * 28 / 5

n = (1.96 * 28 / 5)²

n = 120.47

n = 120

The minimum sample size that will give a margin of error of 5 minutes with 95% confidence is 120.

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Answer:

the answer is A

Step-by-step explanation:

p³-125q³

(p-5q)³

but if expand A

it will give the same p³-125q³

(p-5q)(p²+5pq+25q²)

expanding will give

p3+5p²q+25pq²-5p²q-25pq²-125q³

C.L.T

p³+5p²q-5p²q+25pq²-25pq²-125q³

p³-125q³

The cube root of -1 is -1, and the cube root ∛(-125) is -5.

To find the real nth roots of a number a, we can use the formula:

[tex]√(a) = a^(1/n)[/tex]

For the case where n=3 and a=-125, we have:

[tex]√(-125) = (-125)^(1/3)[/tex]

We can simplify this using the fact that

[tex](-a)^(1/n) = -(a^(1/n)):[/tex]

√(-125) = - (√125)

Now we need to find the cube root of 125. We can factor 125 as 555, so:

√(-125) = - (√125) = - (√(555)) = - (5√5)

Therefore, the real cube root of -125 is -5√5.

The cube root of -125 is the real number x that satisfies the equation

[tex] {x}^{3} [/tex]

= -125.

We can rewrite -125 as -1*5^3, which means we can write the cube root of -125 as ∛(-1)*∛(5^3).

The cube root of -1 is -1, and the cube root of 5^3 is 5, so ∛(-125) = -1*5 = -5.

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The proof of the congruence of the angles ∠QPS and ∠TPR is shown below

Given that

∠QPS ≅ ∠TPR

We have the proof of the congruence of the angles ∠TPS and ∠QPR using the following two column style of proofs

         Statements                                     Reasons

a        ∠QPS ≅ ∠TPR                                Given

b        ∠QPS ≅ ∠TPR                                Given

c        ∠QPS ≅ ∠QPR + ∠RPS                   Addition property

         ∠TPR ≅ ∠TPS + ∠RPS                          

d        ∠TPR ≅ ∠QPR + ∠RPS                    Substitution

e        ∠QPR + ∠RPS ≅ ∠TPS + ∠RPS       Substitution

f         ∠QPS  ≅ ∠TPS                                  Subtract property

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Complete question

Complete the following proof. Given: ∠QPS ≅ ∠TPR

Prove: ∠QPS  ≅ ∠TPS

         Statements                                     Reasons

a        ______________                          ______________

b        ∠QPS ≅ ∠TPR                                ______________

c        ∠QPS ≅ ∠QPR + ∠RPS                   ______________

         ∠TPR ≅ ∠TPS + ∠RPS                          

d        ______________                           Substitution

e        ______________                          ______________

f        ______________                          ______________

Answer:

[tex]\frac{69+2\sqrt{2}}{97}[/tex]

Step-by-step explanation:

(10 + √2)÷(14+ √2)= [tex]\frac{10+\sqrt{2} }{14+\sqrt{2} }[/tex]

                             = [tex]\frac{(10+\sqrt{2}) x (14-\sqrt{2})}{(14+\sqrt{2}) x (14-\sqrt{2}) }[/tex]

                             = [tex]\frac{(10+\sqrt{2}) x (14-\sqrt{2})}{14^{2}-(\sqrt{2}^{2})}[/tex]

                             = [tex]\frac{{(10+\sqrt{2}) x (14-\sqrt{2})}}{196-2}[/tex]

                             = [tex]\frac{140-10\sqrt{2}+14\sqrt{2}-2}{194}[/tex]

                             = [tex]\frac{138+4\sqrt{2}}{194}[/tex]

                             = [tex]\frac{2 x (69+2\sqrt{2})}{194}[/tex]

                             = [tex]\frac{69+2\sqrt{2}}{97}[/tex]

∴ the answer is [tex]\frac{69+2\sqrt{2}}{97}[/tex] in the form [tex]\frac{a+b\sqrt{2}}{c}[/tex]

(10 + √2)÷(14+ √2) can be written in the form (69 + 3√2) ÷ 97, where a = 69, b = 3, and c = 97, and all are integers.

A number that includes zero, positive numbers, and negative numbers, is an integer. Integer's can never be a fraction, a decimal, or a percent, it should be noted.

To write (10 + √2)÷(14+ √2) in the form a+b√2/c, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is 14 - √2.

[(10 + √2) ÷ (14 + √2)] x [(14 - √2) ÷ (14 - √2)]

= [(10 + √2) x (14 - √2)] ÷ [(14 + √2) x (14 - √2)]

= [140 + 8√2 - 2√2 - 2] ÷ [196 - 2]

= (138 + 6√2) ÷ 194

= (69 + 3√2) ÷ 97

Therefore, (10 + √2)÷(14+ √2) can be written in the form (69 + 3√2) ÷ 97, where a = 69, b = 3, and c = 97, and all are integers.

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In this case, there are 2 outcomes for each toss, so there are 2³ = 8 possible outcomes in total.

A tree diagram is a visual tool used to represent a set of possible outcomes or decisions in a systematic way, branching out from a central point.

Here is the tree diagram showing all the possible outcomes for tossing a two-color chip counter three times:

      R         Y

  /        |         \

 R         R         Y

/ \         / \        /

R Y       R Y     R Y

Each branch represents a single toss of the chip counter, with the two possible outcomes represented by the two branches extending from each node.

The outcome of each toss is independent of the outcome of the other tosses, so the total number of possible outcomes is the product of the number of outcomes for each toss.

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Answer:6

Step-by-step explanation:

let's write this as equation,

4x-9=15

4x=15+9=24

x=24/4=6

Lily's error is in the first step, (2+i)^2 ≠ 4 + i^2.

(2+i)^2 = (2+i)(2+i) and you need to FOIL.

(2+i)^2 = (2+i)(2+i)

           = 4 + 2i + 2i + i^2

           = 4 + 4i + (-1)

           = 3 + 4i

Bonus:

If you want to multipy (2+i) by (a+bi) and not end up with a complex number, you'd first FOIL

     (2+i)(a+bi) = 2a + 2bi + ai + bi^2

We know i^2 = -1, so this becomes

                      = 2a + 2bi + ai + b(-1)

                      = 2a + 2bi + ai - b

                      = 2a - b + 2bi + ai

Now, for this not to be complex, we need the imaginary pieces to cancel each other out.  In other words 2bi+ai=0.  For that to happen, 2b + a = 0, or

    2b = -a  or a = -2b

So it would seem that if we pick any b-value and make a = -2b, then we'll end up with a non-complex number.

Let's try b=5, making a = -10

     (2+i)(-10+5i) = -20 + 10i - 10i + 5i^2

The 10i's cancel and 5i^2 = -5, so we're left just with -25.

Step-by-step explanation:

Pool area = L x W = 3 x  * 2x = 6x^2  ft^2

Pool + deck =    (L+10)(W+10) =  (3x+10)(2x+10) = 6x^2 + 50x + 100  ft^2

a) The number of ways to choose three people from a set of eleven is given as follows: 165.

b) The number of ways to choose three women from a set of five women is given as follows: 10.

c) The probability that the group will consist of all women is given as follows: 0.0606 = 6.06%.

The number of different combinations of x objects from a set of n elements is obtained with the formula presented as follows, using factorials.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The number of ways to choose 3 people from a set of 11 is given as follows:

C(11,3) = 11!/(3! x 8!) = 165.

The number of ways to choose 3 women from a set of 5 is given as follows:

C(5,3) = 5!/(2! x 3!) = 10.

Hence the probability that the group will consist of all women is given as follows:

10/165 = 0.0606 = 6.06%.

a group consists of six men and five women.Three people are selected to attend a conference. a) how many ways can three people be selected from this group of eleven?

In how many ways can three women be selected from the five women?

Find the probability that selected group will consist of all women.

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The distances from the barn to the center of each pen are: Horse pen: 47 yards , Cow pen: 50.25 yards , Pig pen: 64.75 yards.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

To solve for the distances from the barn to the center of each pen, we can use the following steps:

Draw a diagram of the situation, labeling the radii of each pen and the distances between the pens.

Use the Pythagorean theorem to solve for the radius of each pen. Let r1 be the radius of the horse pen, r2 be the radius of the cow pen, and r3 be the radius of the pig pen.

For the horse pen:

r1² = (15 + 16)² + (18)²

r1²= 961

r1 = 31 (rounded to the nearest whole number)

For the pig pen:

r3² = (10 + 24.75)² + (18)²

r3²= 1640.06

r3 = 40 (rounded to the nearest whole number)

Use the distance from the barn to the point of tangency of each pen to solve for the distance from the barn to the center of each pen. Let d1 be the distance from the barn to the center of the horse pen, d2 be the distance from the barn to the center of the cow pen, and d3 be the distance from the barn to the center of the pig pen.

For the horse pen:

d1 = 16 + r1

d1 = 16 + 31

d1 = 47 yards

For the cow pen:

d2 = 75 - r2

d2 = 75 - 24.75

d2 = 50.25 yards

For the pig pen:

d3 = 24.75 + r3

d3 = 24.75 + 40

d3 = 64.75 yards

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99.96% of area lies between given value range under the curve.

We can use a standard normal distribution table or a calculator with a normal distribution function to determine the area under the curve of the standard normal distribution between z = -3.5 and z = 3.5. The difference between the area to the left of z = 3.5 and the area to the left of z = -3.5 is the area under the curve between these two numbers.

We determine the region to the left of z = 3.5 and z = -3.5 using a typical normal distribution table. Left of z = 3.5 is an area of 0.9998, whereas left of z = -3.5 is an area of 0.0002. Hence, the region between z = -3.5 and z = 3.5 is as follows:

0.9998 - 0.0002 = 0.9996

To turn this to a percentage, we multiply by 100:

0.9996 x 100 = 99.96%

In other words, 99.96% of the area is within 3.5 standard deviations of the mean, or between z = -3.5 and z = 3.5. This shows that most values in a normal distribution fall within a few standard deviations of the mean and accounts for a sizeable portion of the overall area under the curve.

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Answer: (5x-1)(x+4)

Step-by-step explanation:what she said

The line labelled “W” shows the lowest frequency of sound.

Sound is a type of energy that is created by vibrations and travels through the air in the form of waves. Sound can only be heard when these waves enter the ear and vibrate the eardrum. The frequency of the sound wave determines the pitch of the sound. Sound is also used to communicate and can be manipulated to create music. It can be used to detect events and can travel through water and solid objects.

Frequency is a measure of how often a sound wave repeats itself over time. It is measured in hertz (Hz), which is the number of sound waves per second. The lower the frequency of a sound wave, the lower the pitch of the sound. The line labelled “W” shows the lowest frequency of sound, meaning it has the lowest pitch. The other lines show higher frequencies, meaning they have higher pitches.

The other lines show different frequencies of sound, each with its own pitch. Line “X” has a higher frequency than line “W”, but still lower than the other lines. Line “LO” has a higher frequency than “X”, but lower than the other lines. Line “Y” has a higher frequency than “LO”, but still lower than the other lines. Line “O” has the highest frequency and therefore the highest pitch among all of the lines.

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The dog drinks 112 ounces, or 14 cups, or 7 pints, or 3.5 quarts of water per week.

The dog drinks 16 oz of water every day, so in a week (7 days), the dog will drink:

16 oz/day × 7 days/week = 112 oz/week

To convert ounces to cups, we divide by 8 (since there are 8 fluid ounces in a cup):

112 oz/week ÷ 8 oz/cup = 14 cups/week

To convert ounces to pints, we divide by 16 (since there are 16 fluid ounces in a pint):

112 oz/week ÷ 16 oz/pint = 7 pints/week

To convert ounces to quarts, we divide by 32 (since there are 32 fluid ounces in a quart):

112 oz/week ÷ 32 oz/quart = 3.5 quarts/week

Therefore, the dog drinks 112 ounces, or 14 cups, or 7 pints, or 3.5 quarts of water per week.

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The composition of (x, y) --> (x+4, y-1) and (x, y) --> (x-5, y-5) is (x, y) --> ( x - 1 , y - 6).

What is Composition of transformation ?

Each transformation applied to the prior image is combined into a composition of transformations. The result of a translation is identical to a composition of reflections across parallel lines (twice the distance between the parallel lines)

Given :

1st Translation :  (x, y) --> (x+4, y-1)

It means 4 units to the right and 1 unit down.

2nd Translation : (x, y) --> (x-5, y-5)

It means 5 units to the left and 5 units down.

Now, The composition will be as follows:

(x, y) --> ( x +4 -5 , y - 1 - 5)

(x, y) --> ( x - 1 , y - 6)

It means 1 unit to the left and 6 units down.

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First we need to find the inverse function.  To do this, I'd put f(x) into vertex form:

    [tex]\begin{aligned}f(x) &= (x^2-4x)-6\\[0.5em] &= (x^2-4x+4)-6-4\\[0.5em] &= (x-2)^2-10\\[0.5em]\end{aligned}[/tex]

Now, we can try to invert the function, keeping in mind this is only used when x≥2.

     [tex]\begin{aligned}y&= (x-2)^2-10\\[0.5em]y+10&= (x-2)^2\\[0.5em]\sqrt{y+10}&= x-2\\[0.5em]\sqrt{y+10}+2&= x\\[0.5em]\end{aligned}[/tex]

(The fact that x ≥ 2 allowed us only keep the positive square root on the third line.)

So our inverse function is [tex]f^{-1}(y)=\sqrt{y+10}+2[/tex].

Now, let's find the derivative of this function:

    [tex]\begin{aligned}\dfrac{df^{-1}}{dy}&= \dfrac{1}{2\sqrt{y+10}}\cdot\dfrac{d}{dy}[y+10]\\[0.5em] &= \dfrac{1}{2\sqrt{y+10}}\end{aligned}\\[/tex]

(The chain rule was used, but the derivative of the inside function equals 1.)

So [tex](f^{-1})'(-6) = \dfrac{1}{2\sqrt{-6+10}} = \dfrac{1}{4}[/tex]

Now having done all of that, it did cross my mind that since [tex]f[/tex] and [tex]f^{-1}[/tex] are simply reflections over the line y=x, if we had actually just found f'(4) and then used the reciprocal slope, we'd also get the same answer more quickly.

f'(x) = 2x - 4

when y = -6, then x = 4 (by solving f(x) = -6).

f'(4) = 4, so reflecting that slope of 4/1 over the line y=x, we'd get a slope of 1/4.

If the length of one side of the rhombus is 6m and the radius of the rhombus as 4 m then the area of the rhombus is 24 m².

What is the area of the rhombus?

The Area of a Rhombus = A = ½ × d1 × d2,

Where d1 and d2 are the diagonals of the rhombus.

The area of a rhombus can be found using the formula:

Area = (diagonal 1 x diagonal 2) / 2

or

Area = (base x height) / 2

We are given the length of one side of the rhombus as 6 m, which means the length of the other side is also 6 m since all sides of a rhombus are congruent. We are also given the radius of the rhombus as 4 m, which is half the length of the diagonal.

Using the Pythagorean theorem, we can find the length of the other diagonal:

diagonal 2 = 2(radius) = 2(4) = 8 m

Now we can use the formula to find the area of the rhombus:

Area = (diagonal 1 x diagonal 2) / 2

Area = (6 m x 8 m) / 2

Area = 24 m²

Therefore, the area of the rhombus is 24 m².

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Th function is f(x) = 3.39x²

The cost of the flooring for a dining room is $410.19.

The cost of the flooring for a dining room is $6,072.31.

What is a function?

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

Area of a square = side²

Square with side lengths = x feet.

The tile costs ​= $3.39 per square foot.

a.

The function f for the cost of the flooring.

f(x) = 3.39x²

b.

x =  11 feet

f(11) = 3.39 x 11²

f(11) = 3.39 x 121

f(11) = $410.19

c.

x = 23 feet

f(23) = 3.39 x 23²

f(23) = 3.39 x 529

f(23) = $6,072.31

Therefore,

f(x) = 3.39x²

The cost of the flooring for a dining room is $410.19.

The cost of the flooring for a dining room is $6,072.31.

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Complete Question

Alisa is choosing new tile for the floor in her dining room which is in the shape of a square with side lengths x feet. The tile costs ​$3.39 per square foot.

a. Write the function f for the cost of the flooring.

b. Determine the cost of the flooring if she decides on a dining room with side lengths of 11 ft.

c. Determine the cost of the flooring if she decides on a dining room with side lengths of 23 ft

Ronald spent 50 minutes answering phone calls on Saturday morning. The equation used to find this value is x + 40 = 90.

Let's assume that Ronald spent "x" minutes answering phone calls. We know that he spent a total of 90 minutes volunteering, and 40 minutes delivering flowers. So the time he spent answering phone calls and delivering flowers can be expressed as:

x + 40

We also know that the total time he spent volunteering was 90 minutes. So we can write:

x + 40 = 90

To solve for "x", we can subtract 40 from both sides of the equation:

x + 40 - 40 = 90 - 40

Simplifying:

x = 50

where x represents the amount of time Ronald spent answering phone calls.

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Answer:

2

Step-by-step explanation:

-3x - 12 = 40 - 8x - 21x

-3x + 8x + 21x = 40 + 12

26x = 52

x = 52/26

x = 2

Answer:

x=-14/13, or -1 1/13

Step-by-step explanation:

The measurement of side DE is 3 centimeters.

What is the trapezoids?

A trapezoid is a 2D geometric shape that has four sides, with two sides parallel to each other and two sides non-parallel.

To solve this question, we can use the fact that similar figures have corresponding sides in proportion. Let's label the lengths of the sides of the trapezoids as follows:

First trapezoid: BE = 4.5 cm, BC = 6 cm, CD = 4.5 cm

Second trapezoid: FG = 4 cm, GH = 3 cm, HI = 2 cm, FI = 3 cm

Since the trapezoids are similar, we know that the ratio of corresponding sides is the same. Let's call this ratio k.

For the first trapezoid, the parallel sides are DE and BC, so we have:

k = DE / BC

For the second trapezoid, the parallel sides are FG and HI, so we have:

k = HI / FG

Since the trapezoids are similar, these ratios must be equal:

DE / BC = HI / FG

Substituting in the given lengths, we get:

DE / 6 = 2 / 4

Simplifying, we get:

DE = 3 cm

Therefore, the measurement of side DE is 3 centimeters.

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The box of Cremora is typically rectangular in shape, with six rectangular faces and twelve edges.

Answer:

Step-by-step explanation:

2W+23 is at most 70. So, 2W+23 must be less than or equal to 70.

2W+23≤70

Solve

2W≤47

W≤47/2 or W≤23.5

The solution to the inequality is: x >= 55 or x <= -32.

what is inequality?

An inequality is a mathematical statement that compares two values or expressions and indicates that they are not equal. In other words, an inequality shows the relationship between two values or expressions that are not the same.

To solve the inequality, we need to isolate x on one side of the inequality symbol in each of the two inequalities.

For the first inequality:

3 - 2/5 * x <= -19

Subtracting 3 from both sides, we get:

-2/5 * x <= -22

Dividing both sides by -2/5 (which is the same as multiplying both sides by -5/2), we get:

x >= 55

For the second inequality:

-3/4 * x >= 24

Dividing both sides by -3/4 (which is the same as multiplying both sides by -4/3), we get:

x <= -32

Therefore, the solution to the inequality is:

x >= 55 or x <= -32.

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Answer:

1.5 gallons per minute

Step-by-step explanation:

To solve this question we have to figure out how much water there was in every time scale calculated. To do this we simply have to divide both values...

We don't need to figure out the 0 as no time has passed for water to fill up!

We can see that for each one the answer is 1.5, so our average rate would be 1.5 gallons per minute!

Hope this helps, have a lovely day! :)

The following are the answers:

(e) All the parts are discussed in step 2.

(f) The solution is x= 4 and the verification is shown below

Given the equation y^2  - 23y = 0

We would follow the steps

Given the equation [tex]\frac{16}{6-x} = \frac{56}{x + y}[/tex]

We need to solve this and verify the solution

(e) (i) Substitute y = 23 on the left-hand side of the given equation.

23^2 -23(23)  = 0

0 = 0

Since the left-hand side is equal to the right-hand side, hence y= 23 is the solution of the given equation, but it is not the only solution.

(ii) Complete solution of the given equation:

y^2 -23y = 0

y(y-23) =0

y= 0

and y -23 = 0

y = 0, 23

Hence the complete solution of the given equation is y= 0, 23

EXPLANATION

If a.b = 0, then a= 0, b=0

.

(iii) The solution is inadequate because we can cancel the term only if it non zero.

Also, the given equation has power 2, but the student is getting only a single value, which is incorrect.

Solve the given equation as follows:

[tex]\frac{16}{6-x} = \frac{56}{x + 3}[/tex]

16 (x +3)= 56(6-x)

= 2x + 6= 42- 7x

9x = 36

x= 4

Hence the required solution is x= 4

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The probability that both Raju and Akash will solve any random problem given to them after sufficient time is 1/4.

The probability that Raju will solve a random problem given to him is 1/3 and the probability that Akash will solve the same problem is 3/4. We can use the multiplication rule of probability to find the probability that both of them will solve any random problem given to them after sufficient time.

According to the multiplication rule of probability, the probability of two independent events A and B occurring together is the product of their individual probabilities:

P(A and B) = P(A) × P(B)

In this case, the event "Raju solves a problem" is independent of the event "Akash solves a problem". Therefore, the probability that both Raju and Akash will solve a random problem given to them after sufficient time is:

P(Raju and Akash) = P(Raju) × P(Akash)

= 1/3 × 3/4

= 1/4

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