A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (see figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge in order to empty the following amounts?

Since x is less than -6, we use the first equation to calculate h(-6):

h(-6) = 4(-6) - 20 = -24

A function is a mathematical concept that assigns to each input value (or "argument") exactly one output value (or "image"). In other words, a function is a rule that assigns a unique output for each input value. The set of input values is called the domain of the function, and the set of output values is called the range. A function can be represented graphically as a curve, or analytically as a formula. Functions play a central role in many areas of mathematics, science, and engineering.

For x < -6, h(x) = -4x - 20. So, h(-6) = -4(-6) - 20 = 24 - 20 = 4.

For -6 ≤ x < 1, h(x) = x + 5. So, h(0) = 0 + 5 = 5.

For x ≥ 1, h(x) = 1. So, h(1) = h(7) = 1.

So, the values are:

h(-6) = 4, h(0) = 5, h(1) = h(7) = 1.

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Morse's monthly budget has a surplus of $3,860 (rounded to the nearest whole number).

From the we are to calculate Morse's monthly budget.

To determine Morse's monthly budget, we need to first calculate his total annual expenses:

Total Annual Expenses = Fixed Expenses + Living Expenses + Annual Expenses

= $2,117 + $489 + $475

= $3,081

Then, we can calculate his monthly budget by dividing his annual net income by 12:

Monthly Budget = Annual Net Income / 12

= $49,397 / 12

= $4,116.42

Now, we can determine Morse's monthly budget by subtracting his total monthly expenses from his monthly net income:

Monthly Budget = Monthly Net Income - Monthly Expenses

Monthly Net Income = Annual Net Income / 12 = $49,397 / 12 = $4,116.42 (rounded to the nearest cent)

Monthly Expenses = Total Annual Expenses / 12 = $3,081 / 12 = $256.75 (rounded to the nearest cent)

Monthly Budget = $4,116.42 - $256.75 = $3,859.67

Hence, Morse's monthly budget has a surplus of $3,860

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The solution to the parts of the question with regards to the quadratic equation are;

A quadratic equation is an equation of the form f(x) = a·x² + b·x + c

The discriminant, D, of a quadratic equation, f(x) = a·x² + b·x + c, can be obtained using the expression;

D = b² - 4 × a × c

The specified quadratic function is; -4·z² - 3·z + 5 = 0

The discriminant, D of the above quadratic expression is therefore;

D = (-3)² - 4 × (-4) × 5 = 89

The discriminant is larger than zero, therefore, the quadratic expression has two solutions.

The two method to be used to find the specific solution are;

Quadratic Formula;

The solutions of the quadratic equation based on the quadratic formula are;

z = (-(-3) ± √((-4)² - 4 × (-4) × 5))/(2 × (-4))

z = (3 ± √(89))/(-8)

Completing the Square

The completing the square method can be used as follows;

-4·z² - 3·z + 5 = 0

z² + (3/4)·z - 5/4 = 0

z² + (3/4)·z  = 5/4

z² + (3/4)·z + ((3/4)/2)² = 5/4 + ((3/4)/2)²

z² + (3/4)·z + (3/8)² = 5/4 + (3/8)²

(z + (3/8))² = 5/4 + (3/8)²

z + (3/8) = ±√((5/4) + (3/8)²)

z = ±√(5/4 + (3/8)²) - (3/8)

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The  answer of this queation :∫ (4 + √x + x/x) dx = 4x + 2/3 x^(3/2) + x + C

where C = C1 + C2 + C3 is the constant of integration for the entire expression.

eparate integrals:

∫ 4 dx + ∫√x dx + ∫ x/x dx

The first two integrals can be easily integrated as follows:

∫ 4 dx = 4x + C1, where C1 is a constant of integration.

∫√x dx = 2/3 x^(3/2) + C2, where C2 is a constant of integration.

For the third integral, note that x/x simplifies to 1 for all nonzero x.

∫ x/x dx = ∫ 1 dx = x + C3, where C3 is a constant of integration.

Putting it all together, we have:

∫ (4 + √x + x/x) dx = 4x + 2/3 x^(3/2) + x + C

where the integration constant for the entire statement is C = C1 + C2 + C3.

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A tree with a height of 6.2 ft exists 3 standard deviations above the mean.

The normal distribution represents a symmetrical display of data around its mean value, with the standard deviation serving as the determinant of the curve's breadth. The "bell curve" is used to visually represent it.

The normal distribution, sometimes referred to as the Gaussian distribution, is a probability distribution that is symmetric about the mean and demonstrates that data that exists closer to the mean exists more likely to even than data that exists farther from the mean. The normal distribution is depicted graphically as a "bell curve."

The majority of data points in a continuous probability distribution known as a "normal distribution" cluster around the middle of the range, while the remaining data points taper off symmetrically toward either extreme. The mean of the distribution is another name for the center of the range.

It exists said that an X value exists found Z standard deviations from the mean if:

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

Given:

[tex]$$\begin{aligned}& \mu=5 \mathrm{ft} \\& \sigma=0.4 \mathrm{ft}\end{aligned}$$[/tex]

We have four different values of X and we must calculate the Z-score for each

For X = 5.4 ft

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

substitute the values in the above equation, we get

[tex]$ Z=\frac{5.4-5}{0.4}=1\end{aligned}$$[/tex]

A tree with a height of 5.4 ft exists 1 standard deviation above the mean

First Option: False

For X = 4.6 ft

[tex]$Z=\frac{4.6-5}{0.4}=-1\end{aligned}$$[/tex]

A tree with a height of 4.6ft exists 1 standard deviation below the mean

Second Option: False

For X = 5.8 ft

[tex]$Z=\frac{5.8-5}{0.4}=2\end{aligned}$$[/tex]

A tree with a height of 5.8 ft exists  2 standard deviation above the mean

Third Option: False

For X = 6.2 ft

[tex]$ Z=\frac{6.2-5}{0.4}=3\end{aligned}$$[/tex]

A tree with a height of 6.2 ft exists 3 standard deviations above the mean.

Therefore, the correct answer is option d) A tree with a height of 6.2 ft is 3 standard deviations above the mean.

The complete question is;

The heights of a certain type of tree are approximately normally distributed with a mean height y = 5 ft and a standard deviation = 0.4 ft. Which statement must be true?

a) A tree with a height of 5.4 ft is 1 standard deviation below the mean

b) A tree with a height of 4.6 ft is 1 standard deviation above the mean.

c) A tree with a height of 5.8 ft is 2.5 standard deviations above the mean

d) A tree with a height of 6.2 ft is 3 standard deviations above the mean.

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Nancy collected 5 5/6 cans of bags which is 7 times more than her friend.

A fraction is written in the form of p/q, where q ≠ 0.

Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.

Given, Nancy collected 7 times as many bags of can and her friend collected 5/6 of a bag.

Therefore, The amount of bags of cans Nancy collected can be obtained by multiplying 7 by the number of bags of can collected by her friend which is,

= (5/6)×7 bags.

= 35/6 bags.

= 5 5/6 can of bags.

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The statement "This door will not open unless a security code is entered" can be expressed in if-then form as "If a security code is not entered, then the door will not open."

This is because the statement "unless" indicates a necessary condition. In other words, the security code must be entered in order for the door to open, and if the security code is not entered, then the door will not open. Therefore, the correct option is: If a security code is not entered, then the door will not open.

The capacity to use numerical data to recognise patterns, solve problems without a pre-existing formula, understand graphs, and reach reasonable conclusions when faced with numerical evidence is known as mathematical reasoning.

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Answer:

76 can be written as the product of 2 and 38. It is given as: √ 76 = √ (2 × 38) 2 is not a perfect square. Hence, it stays within the root sign. 38 can be shown as 2 × 19. The simplified radical form of the square root of 76 is 2 √ 19.

The given list of functions can be arranged in ascending order of growth rate as follows: g1(n), g5(n), g3(n), g4(n), g2(n), g6(n), and g7(n).

The Big O notation describes the upper bound of a function's growth rate. In other words, it represents the maximum amount of time or space that a function requires to complete its operations.

Using this concept, we can arrange the given list of functions in ascending order of growth rate as follows:

g1(n) = √2 log n: This function has a growth rate of O(log n), which is less than the growth rates of all other functions in the list.

g5(n) = n log n: This function has a growth rate of O(n log n), which is greater than the growth rate of g1(n), but less than the growth rates of all other functions in the list.

g3(n) = n 4/3: This function has a growth rate of O(n 4/3), which is greater than the growth rates of g1(n) and g5(n), but less than the growth rates of all other functions in the list.

g4(n) = n(log n)3: This function has a growth rate of O(n(log n)3), which is greater than the growth rates of g1(n), g5(n), and g3(n), but less than the growth rates of all other functions in the list.

g2(n) = 2n: This function has a growth rate of O(2n), which is greater than the growth rates of g1(n), g5(n), g3(n), and g4(n), but less than the growth rates of g6(n) and g7(n).

g6(n) = 22 n: This function has a growth rate of O(2n), which is greater than the growth rates of g1(n), g5(n), g3(n), g4(n), and g2(n), but less than the growth rate of g7(n).

g7(n) = 2n2: This function has a growth rate of O(2n2), which is greater than the growth rates of all other functions in the list.

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Complete Question:

Arrange the following list of functions in ascending order of growth rate, i.e. if the function g(n) immediately follows f(n) in your list then, it should be the case that f(n) = O(g(n)).

g1(n) = √2 log n

g2(n) = 2n

g3(n) = n 4/3

g4(n) = n(log n)3

g5(n) = n log n

g6(n) = 22 n

g7(n) = 2n2

Answer:

x=41,y=15

Step-by-step explanation:

The given system of equations is:

x = 41

x + 2y = 71

We can use substitution to solve for y.

Starting with the first equation:

x = 41

We can substitute the value of x into the second equation:

x + 2y = 71

41 + 2y = 71

2y = 71 - 41

2y = 30

Finally, we can solve for y by dividing both sides of the equation by 2:

y = 30 / 2

y = 15

x=41,y=15.

The measure of each of the angles in the triangle given is 48°, 64° and 68°.

Ratio defines the relationship between two quantities where it tells how much one quantity is contained in the other.

The ratio of a and b is denoted as a : b, which means that a parts of a quantity is corresponding to b parts of another quantity.

Given that,

Ratio of measures of two angles of a triangle = 3 : 4

Let x be a number such that,

Measure of smaller angle =3x

Measure of second angle = 4x

Measure of third angle is 20° more than the measure of the smaller of the first two angles.

Measure of third angle = 3x + 20

We know that sum of interior angles of a triangle = 180°

3x + 4x + (3x + 20) = 180

10x + 20 = 180

10x = 160

x = 16

Measure of smaller angle = 3x = 48°

Measure of second angle = 4x = 64°

Measure of third angle = 3x + 20 = 68°

Hence the measures of each of the angle in triangle is 48°, 64° and 68°.

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The volume of the solid generated by revolving the region about the y-axis is approximately 0.200 cubic units.  

To use the shell method to find the volume of the solid generated by revolving the region bounded by the curves [tex]$y=1$[/tex], [tex]$y=\frac{1}{2\pi e^{x^2/7}}$[/tex], [tex]$x=0$[/tex], and [tex]$x=1$[/tex] about the y-axis, we need to integrate along the x-axis.

The basic idea of the shell method is to take a vertical strip of width [tex]$dx$[/tex]and height [tex]$f(x)$[/tex] and revolve it about the y-axis to generate a thin shell of thickness [tex]$dx$[/tex] and radius x.

The volume of the solid is then given by the integral:

[tex]$$V = \int_{x=0}^{x=1} 2\pi x f(x) dx $$[/tex]

where [tex]$f(x)$[/tex] is the height of the shell at the position [tex]$x$[/tex]. In this case,

[tex]$f(x) =[/tex] [tex]1 - \frac{1}{2\pi e^{x^2/7}}$.[/tex]

So, we have:

[tex]$$V = \int_{x=0}^{x=1} 2\pi x \left(1 - \frac{1}{2\pi e^{x^2/7}}\right) dx $$[/tex]

Now, we can evaluate this integral using integration by substitution.

Let [tex]$u=x^2/7$[/tex], so [tex]$du/dx = 2x/7$[/tex] and [tex]$x,dx = 7/2,du$[/tex]. The integral becomes:

[tex]$$V = \int_{u=0}^{u=1/7} \frac{2\pi}{7} e^{-u} (7/2) du = \pi\int_{0}^{1/7} e^{-u} du$$[/tex]

Evaluating this integral gives:

[tex]$$V = \pi\left[-e^{-u}\right]_{0}^{1/7} = \pi\left(1 - e^{-1/7}\right) \approx \boxed{0.200}$$[/tex]

Therefore, the volume of the solid generated by revolving the region about the y-axis is approximately 0.200 cubic units.

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The cost of each container is $0.18.

In mathematics, volume is the space taken by an object. Volume is a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units. The definition of length is interrelated with volume.

here, we have,

from the given figure we get,

container's length = 6in

width = 2.5in

height = 8in

volume of the container =120 in^3

each container will be made from material that costs $0.0015 per cubic inch.

so, the costs of each container = $0.0015* 120

                                                    = $0.18

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The distance between the rests in the new map is 0.825 inches.

We know that the original scale is:

2 in = 33mi

or:

1 in = (33mi)/2

1in = 16.5 mi

And on a highway, there are 7 rests in 9 inches.

First, we transform these 9 inches to miles

9 in = 9*(16.5 mi) = 148.5 mi

If the 7 rests are evenly divided in that distance, the distance between each rest is:

148.5mi/6 = 24.75mi

(we divide by 6 because one rest is at each end, so there are 6 even spaces between the two ends)

Now, in the new map the scale is:

1 inch = 30mi

Then the distance between the rests in the new map is:

d = 24.75/30 inches

d = 0.825 inches.

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Answer:

the answer to your question is x=3.

Step-by-step explanation:

hope this helps.

Step-by-step explanation:

arrange ff fraction 5/6,8/9,23

Answer:

the value of the original number (n) is 25.

Step-by-step explanation:

Let's use algebra to solve the problem.

We are told that a number (n) is increased by 10 and the result is doubled, which gives:

2(n + 10)

We are also told that this result is 5 less than three times the original number (n), which gives:

2(n + 10) = 3n - 5

Expanding the left side of the equation:

2n + 20 = 3n - 5

Subtracting 2n from both sides:

20 = n - 5

Adding 5 to both sides:

25 = n

Therefore, the value of the original number (n) is 25.

The missing sides of the triangles are given below.

Length is defined as the measurement of distance of an object from one end to the other.

To find the missing sides of the given triangles.

Question 1 :

In the triangle, consider sin 90° = [tex]\frac{opposite side}{hypotenuse side}[/tex]

                                               1 = [tex]\frac{13}{x}[/tex]

                                              ⇒ x = 13

Which is the length of the missing side of the triangle.

Question 2:

In this triangle, consider degree 63° we have to find the  length of hypotenuse side, then,

sin 63° =[tex]\frac{opposite side }{hypotenuse side}[/tex] = [tex]\frac{18}{x}[/tex]

0.89 x = 18

⇒ x = 18/0.22 = 20.22.

Length of the hypotenuse side is 20.22 cm.

Question 4:

In this triangle, consider cos function.

sin 18°= [tex]\frac{opposite side}{hypotenuse}[/tex] = x/11

0.3090*11 = x

x = 3.399 = 3.4 inches.

Question 5:

consider sin angle.

sin 90° = [tex]\frac{opposite side}{hypotenuse}[/tex] = x/21

⇒x = 21 yard.

Question: 6

For this triangle we consider, tan functions.

tan 43° = [tex]\frac{opposite side}{adjacent side}[/tex] = x/23

⇒ x =21.4 mm.

Question 7:

For this triangle , we consider sin function.

sin 33° = 9/x

⇒0.5446 *x=9/0.5446

⇒x = 16.5km

Question 8:

For this triangle we have to choose tan function,

tan67 ° = opposite side/ adjacent side

            = 17/x

⇒x = 7m

Question 9:

For this triangle, we take sin function.

sin 90°= opposite side/hypotenuse

        1   = x/19

⇒x = 19m

Question 10:

For this triangle we consider sin function,

sin 90°= 26/x

⇒x = 26 feet.

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The composite function of f(x) and g(x) is given as follows:

f(g(x)) = 6x² - 10x - 21.

The composite function of f(x) and g(x) is given by the following rule:

(f ∘ g)(x) = f(g(x)).

It means that the output of the inside function serves as the input for the outside function.

The function g(x) in this problem is given as follows:

g(x) = 3x² - 5x - 7.

Hence, for the composite function in this problem, the lone instance of x in f(x) is replaced by 3x² - 5x - 7, as follows:

f(g(x)) = f(3x² - 5x - 7) = 2(3x² - 5x - 7) - 7 = 6x² - 10x - 21.

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Probability that the selected card bears a Perfect square is 1/10 and the probability that the selected card bears a two digit number is 81/90.

It is a branch of mathematics that deals with the occurrence of a random event.

Given that a box contains cards, number 1 to 90.

A card is drawn at random from the box.

We have to find the the probability that the selected card bears a two-digit number.

In 90 cards, from 1 to 9 are single digit numbers which are 9 in number.

The remaining 81 are two-digit numbers (10 to 90). So the probability of selecting a two-digit number is

81/90

(ii) There are 9 perfect squares between 1 and 90, namely 1, 4, 9, 16, 25, 36, 49, 64, and 81.

So the probability of selecting a perfect square number is:

P(Perfect square) = 9/90 = 1/10

Hence, the probability that the selected card bears a two digit number is 81/90 and probability that the selected card bears a Perfect square is 1/10.

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a) The distribution of Z is given by: P(Z = 1) = 1 - F(2T, T), P(Z = 0) = F(2T, T)

b) X and Z are not independent,  Y and Z are not independent, and pair (X, Y) and Z are also not independent.

c) )X and Y are not independently existent.

a) The distribution of Z can be determined by finding the probability that S > 2T. Let F(s,t) be the joint cumulative distribution function of S and T. The probability that S > 2T is given by:

P(Z = 1) = P(S > 2T) = ∫∫_{2t < s} f(s,t) ds dt = 1 - F(2T, T)

Since T is nonnegative and has a continuous distribution, the cumulative distribution function F(2T, T) is also continuous and ranges from 0 to 1. Therefore, the distribution of Z is given by:

P(Z = 1) = 1 - F(2T, T), P(Z = 0) = F(2T, T)

b) X and Z are not independent, since the value of X affects the probability that S > 2T. For example, if X = x, then T >= x/2, so the value of Z depends on the value of X. Similarly, Y and Z are not independent, since the value of Y affects the probability that S > 2T. For example, if Y = y, then T <= y/2, so the value of Z depends on the value of Y.

The pair (X, Y) and Z are also not independent since the joint distribution of (X, Y) affects the probability that S > 2T. For example, if (X, Y) = (x, y), then T >= x/2 and T <= y/2, so the value of Z depends on the values of X and Y.

c) X and Y are not independent, since the value of X affects the value of Y. For example, if X = x, then Y >= x, so the value of Y depends on the value of X.

The complete question is:-

(Order statistics and independence) Let X be the minimum and Y the maximum of two independent, nonnegative random variables S and T with common continuous density f. Let Z denote the indicator function of the event (S > 2T). a) What is the distribution of Z? b) Are X and Z independent? Are Y and Z independent? Are (X, Y) and Z independent? c) Is X independent of Y?

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The statement as an algebrai equation is x + 1/3 = 3/4

From the question, we have the following parameters that can be used in our computation:

The sum of X and one third is three fourths

In mathematics and algebra, we have

One third = 1/3

Three fourths = 3/4

So, the statement becomes

The sum of X and 1/3 is 3/4

Express as a summation equation

This gives

x + 1/3 = 3/4

Hence, the equation is x + 1/3 = 3/4

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Answer: draw a number line and plot the 0, 1 , and 2

Explanation:

the volume of the solid is x²/2R.

Let x be the length of the base of the rectangle.

The volume of the solid is given by:

V = ∫R 2x dx

= 2∫R x dx

= 2[x²/2]∫R dx

= x²/2 ∫R dx

= x²/2 (R - 0)

= x²/2 R

The volume of the solid is given by the integral of the cross sectional area of the solid. The cross sectional area is a rectangle whose base is x and the height is twice the length of the base. Therefore, the area of the cross section is 2x. The volume of the solid is calculated by integrating the area over the range of the variable, which in this case is R. The integral of 2x over the range R is 2x times R (2x*R). This can be simplified to x squared over two times R (x^2/2*R). Therefore, the volume of the solid is x squared over two times R.

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Answer:

x ≤ 1.5

Step-by-step explanation:

Brett can pour at most 1.5 more gallons of water into his fish tank. This is because the fish tank can only hold 10.5 gallons of water and Brett has already poured 9 gallons into the tank. So, the maximum amount of water he can pour is the remaining 1.5 gallons.

Answer:

djsushf sjsushsjd sjdusbs

Step-by-step explanation:

iahsjsidjdjdudjdbdjdid

D. False. The combined effect of two linear transformations is always linear because multiplying two linear functions together will result in a function which is also linear.

The statement says when two linear transformations are performed one after another, the combined effect may not always be a linear transformation.

This is false,  the combined effect of two linear transformations is always linear because multiplying two linear functions together will result in a function which is also linear.

A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. A linear transformation is also known as a linear operator or map.

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The solution is, their ages are 13 & 39 yrs., when the sum of the ages of a man and his son is equal to twice the difference of that ages, the product of their ages is 507.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

The sum of the ages of a man and his son is equal to twice the difference of that ages,

the product of their ages is 507.

let, their ages are, a &b

now, The sum of the ages of a man and his son is equal to twice the difference of that ages,

so, we get,

a+b = 2(a-b)...(1)

and, ab = 507....(2)

we get,

from (1) we get,

solving both side,

2a - a = b + 2b

or, a = 3b

now, putting the value of a in (2),

from (2) we get,

so, 3b^2 = 507

solving we get,

b = 13

a= 39

Hence, The solution is, their ages are 13 & 39 yrs.

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The percentage discount is 95 percent

A percentage discount is a reduction in price that is expressed as a percentage of the original price.  Percentage discounts are commonly used in retail sales and promotions to incentivize customers to make purchases.

To determine the percentage discount of the orange, we can find the original price.

0.75 * 7 = 5.25

This is the cost of orange

The total amount charged = 4.30

The discount = 5.25 - 4.30 = 0.95

The percentage discount will be;

percentage discount = 0.95 * 100 = 95%

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