The ages of all ten Mathematics teachers at a school are recorded below. 31, 43, 25, 37, 28, 22, 25, 32, 31, 36 Two new Mathematics teachers then join the school. When their ages are added to this list: the range increases by 5 the mean increases by 1 the median remains unchanged the mode remains the same. How old are the two new Mathematics teachers? You must show your working.​

Answer:

z= 63 degrees

x≈16

Step-by-step explanation:

90+27=117

180-117=63 (bc all angles in a triangle have to add up to 180)

so the angle opposite from the 27 is 63 degrees.

Because of opposite interior angles, z=63.

sin 63= [tex]\frac{14}{x}[/tex]

xsin63=14

x=[tex]\frac{14}{sin63}[/tex]

x≈16

Answer:

To round 5425.92 to the nearest whole number, we need to look at the digit in the ones place, which is 5. Since 5 is greater than or equal to 5, we need to round up the nearest whole number. Therefore, 5425.92 rounded to the nearest whole number is 5426.

An exponential function with a vertical stretch by a factor of 2 and a translation left 1 unit and up 2 units is given as follows:

y = 2(2)^(x + 1) + 2.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

A vertical stretch by a factor of k means that the definition of the function is multiplied by k.

The parent function in the context of this problem is given as follows:

y = 2^x.

Hence the exponential function with a vertical stretch by a factor of 2 and a translation left 1 unit and up 2 units is given as follows:

y = 2(2)^(x + 1) + 2.

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Answer:

A

Step-by-step explanation:

These are cross angles

As per the information provided, the answer to all the parts in the question will be as follows:

a. To solve the LP problem by ignoring the integer constraints in Excel using Solver, we can follow the steps below:

Enter the objective function and the constraints in a new Excel worksheet: Objective function: Maximize 15x1 + 2x2 Constraints: 7x1 + x2 <= 23 3x1 - x2 <= 5

Open the Solver add-in by clicking on Data -> Solver in the Excel menu.

Set the objective function to maximize and set the variable cells to x1 and x2. Set the constraints by clicking on Add in the Solver Parameters dialog box.

Set the Solver options to "Assume Linear Model" and "Make Unconstrained Variables Non-Negative". Click Solve. The solution to the LP problem is x1=3, x2=2.714, with an optimal objective function value of 51.714.

b. If we round up fractions greater than or equal to 1/2, the solution becomes x1=3, x2=3, with an objective function value of 51. This is not the optimal integer solution, as we will see in part d.

c. If we round down all fractions, the solution becomes x1=2, x2=2, with an objective function value of 34. This is not the optimal integer solution either, as we will see in part d.

d. To solve the ILP problem in Excel using Solver, we can follow the steps below:

Open the Solver add-in by clicking on Data -> Solver in the Excel menu. Set the objective function to maximize and set the variable cells to x1 and x2. Set the constraints by clicking on Add in the Solver Parameters dialog box. Set the Solver options to "Assume Linear Model" and "Make Unconstrained Variables Non-Negative". Add integer constraints by clicking on Add in the Solver Parameters dialog box, and setting the integer constraints for x1 and x2. Click Solve.

The solution to the LP problem is x1=2, x2=3, with an optimal objective function value of 48.

As we can see, the optimal objective function value for the LP problem (48) is lower than that for the LP problem (51.714), regardless of rounding up or down.

e. The optimal objective function value for the ILP problem is always less than or equal to the corresponding LP's optimal objective function value because the LP problem allows fractional solutions, while the ILP problem only allows integer solutions. Introducing additional constraints that restrict the variables to integers can only reduce the feasible solution space, and thus lead to a lower optimal objective function value. The LP and ILP problems would be equal if the optimal solution for the LP problem happens to be an integer solution.

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Note that the full question is:

Given the following all-integer linear program: (COMPLETE YOUR SOLUTION IN EXCEL USING SOLVER AND UPLOAD YOUR FILE. BE SURE THAT EACH WORKSHEET IN THE EXCEL FILE CORRESPONDS TO EACH QUESTION BELOW ) Max 15x1 + 2x2 s. t. 7x1 + x2 < 23 3x1 - x2 < 5 x1, x2 > 0 and integer a. Solve the problem (using SOLVER) as an LP, ignoring the integer constraints.

b. What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution? c. What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain. d. Show that the optimal objective function value for the ILP is lower than that for the optimal LP (Eg. Resolve original problem using SOLVER with the Integer requirement). e. Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal?

The measure of the angle ABC, when A, B and C lie on circumference is 40 degrees,

Given that,

Central angle is the angle that has its vertex in the center of the circumference and the sides are radii of it

the center of the circle is point 'B'.

three points 'A', 'C', and 'D' lies on the circumference of the

circle.

And angle ADC = 20°.

So, the angle ABC = 40°.

Because the angle made from the center of the circle is twice

the angle made by the three points on the circumference

from the same base.

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( To find the lowest common multiple of any number you should find the prime factorization of 418)

(Attached)

418 = 2 × 11 × 19

In an trial, there's a 1/6 chance that the coming gravestone chosen from the bag will be green.

Experimental probability, occasionally appertained to as empirical probability, is grounded on valid trials and sufficient attestation of the circumstance of events. A number of real tests are carried out to ascertain whether any event will do. Random trials are tests that have an changeable outgrowth. similar trials' results are noway certain. To establish their possibility, arbitrary trials are conducted constantly. Each repeating of an trial which is done a destined number of times is appertained to as a trial. The experimental probability formula is defined as follows in mathematics

Probability of an Event P( E) = Number of times an event occurs /Total number of trials.

According to question,

The experimental liability that a green marble will be the coming one drawn from the bag is equal to the total number of green marbles divided by the total number of marbles.

2/( 2 + 10)

= 2/12

= 1/6

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The equation for the new function F(t) is F(t) = (27/5)t + 23. This function converts the temperature in Celsius as a function of time t to the temperature in Fahrenheit.

We are given the conversion formula F(t) = (9/5)C(t) + 32, where F(t) represents the temperature in Fahrenheit, and C(t) represents the temperature in Celsius as a function of time t. To find the equation for the new function F(t), follow these steps:

1. Identify the given equation for C(t). This should be provided in the problem statement.
2. Substitute the equation for C(t) into the conversion formula F(t) = (9/5)C(t) + 32.

For example, let's assume the equation for C(t) is C(t) = 3t - 5.

Step 1: We have the equation C(t) = 3t - 5.

Step 2: Substitute C(t) into the conversion formula:

F(t) = (9/5)(3t - 5) + 32

Now, simplify the equation:

F(t) = 9/5(3t) - 9/5(5) + 32
F(t) = (27/5)t - 9 + 32
F(t) = (27/5)t + 23

So, the equation for the new function F(t) is F(t) = (27/5)t + 23. This function converts the temperature in Celsius as a function of time t to the temperature in Fahrenheit.

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Answer:

A. The first step is to perform the multiplication operation, which is 8*2.

B. The second step is to perform the addition and subtraction operations from left to right, which gives us the simplified expression of 6 + 20 + 16.

C. The value of the expression is 42.

Step-by-step explanation:

For the given question, the first step in evaluating the expression 6 + (24 - 4) + 8 x 2 is to perform any multiplication or division operations , since they take precedence over addition and subtraction. In this case, 8 x 2 equals 16, so we can simplify the expression to 6 + (24 - 4) + 16.

The second step is to perform any addition or subtraction operations, working from left to right. In this case, 24 - 4 equals 20, so we can further simplify the expression to 6 + 20 + 16.

Finally, we can add the remaining numbers to get the value of the expression: 6 + 20 + 16 = 42. Therefore, the value of the expression is 42.

So, the probability of picking a 4 and then picking an even number is 15/28.

To find probability, you need to follow these steps:

Identify the total number of possible outcomes of an event. Let's call this number "n."

Identify the number of favorable outcomes, which are the outcomes that you're interested in. Let's call this number "f."

Calculate the probability of the event as the ratio of the number of favorable outcomes to the total number of outcomes:

Probability = f / n

For example, if you're flipping a fair coin, there are two possible outcomes: heads or tails. The total number of outcomes is 2, and the number of favorable outcomes is 1 (either heads or tails). Therefore, the probability of getting heads is:

Probability of heads = 1 / 2 = 0.5.

There are four possible outcomes for the first card: 4, 5, 6, or 7. If we pick a 4 on the first draw, then there are three even numbers left to choose from: 6, 4, and 6. If we pick a non-4 number on the first draw, then there are four even numbers left to choose from: 4, 6, 4, and 6.

Therefore, the probability of picking a 4 and then picking an even number is:

[tex](1/4) * (3/7) + (3/4) * (4/7)[/tex]

[tex]= (3/28) + (3/7)[/tex]

[tex]= (3/28) + (12/28)[/tex]

[tex]= 15/28[/tex]

So, the probability of picking a 4 and then picking an even number is 15/28.

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The required rate of  increasing in volume of a sphere when diameter is 80 mm is equals to 25600π mm^3/s

let 'V' be the volume of the volume of a sphere

'r' be the radius of the sphere.

The formula for the volume of a sphere in terms of its radius,

V = (4/3) × π × r^3

Taking the derivative with respect to time, we get,

dV/dt = 4× π × r^2 × (dr/dt)

where dV/dt is the rate of change of the volume,

dr/dt is the rate of change of the radius,

And π is a constant.

The rate of change of the radius dr/dt is 4 mm/s.

Calculate the rate of change of the volume when the diameter is 80 mm.

Diameter = 2 (radius)

This means the radius is 40 mm.

Substituting the values in the formula above, we get,

⇒dV/dt = 4 × π × (40 mm)^2 ×(4 mm/s)

Simplifying this expression, we get,

⇒ dV/dt = 25600π mm^3/s

Therefore, the volume of the sphere is increasing at a rate of 25600π mm^3/s when the diameter is 80 mm.

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When we substitute the figures for x and y, we will arrive at a final figure of -150.

The equivalent expressions for the properties of the expression are as follows:

5 (-6x) + 5 (2y)

5.-6x + 5.2y

-30x + 10y

The question tells us to substitute x for 4 and y for -3 in the equation. To do this, we will have the following:

5 (-6 *4 + 2 * -3)

= 5 (-24 + -6)

= 5 (-24 - 6)

= 5 ( -30)

= -150

So after substituting the letter, we will have -150 as the result.

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For diameter 12 inches the would take approximately 904.78 cubic inches of air to inflate the soccer ball.

Diameter is a straight line segment that passes thrοugh the center οf a circle οr sphere, cοnnecting twο pοints οn the circumference. It is the lοngest chοrd οf the circle οr sphere and its length is twice the radius.

Tο find οut hοw much air is needed tο inflate the sοccer ball, we need tο first calculate the vοlume οf the ball. The fοrmula fοr the vοlume οf a sphere is:

V = (4/3)π[tex]r^3[/tex]

where r is the radius of the sphere.

Since the diameter of the soccer ball is 12 inches, the radius is half of that, or 6 inches. We can substitute this value into the formula and simplify:

V = (4/3)π([tex]6^3[/tex]) = 904.78 cubic inches

This is the volume of the fully inflated soccer ball. If the ball is currently deflated, we need to add enough air to bring its volume up to 904.78 cubic inches.

The amount of air needed will depend on the pressure of the air being used to inflate the ball. If we assume that the pressure is constant, we can use the ideal gas law to calculate the volume of air needed:

PV = nRT

where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature.

Since we are assuming that the pressure and temperature are constant, we can simplify the formula to:

V = (n/R)P

where n/R is a constant for a given amount of gas, and P is the pressure.

Without knowing the pressure of the air being used, we cannot calculate the exact amount of air needed. However, we can assume a standard pressure of 14.7 pounds per square inch (psi) and use this to calculate the volume of air needed.

Assuming a pressure of 14.7 psi, we can calculate the volume of air needed as follows:

V = (n/R)P = (1/14.7)(14.7) = 1 cubic inch

Therefore, it would take approximately 904.78 cubic inches of air to inflate the soccer ball. However, the exact amount of air needed will depend on the pressure of the air being used.

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Thus, the line of best fit for the given speed x (miles per hour)  and their corresponding Braking distance y (feet) is drawn.

A mathematical idea that correlates points spread throughout a graph is called the line of best fit. The optimum technique to define the association between the dots is determined using scatter data in this type of linear regression.

Given table is:

Speed, x (miles per hour) -     20    30     40    50     60

Braking distance, y (feet) -     20    45     80    125    180

Plot the (x,y) coordinates on the graph:

(20, 20) , (30, 45), (40, 80), (50, 125), (60, 180)

line of best fit defines the line, which is the average of the values covering the given data.

For the given data: line of best fit is drawn.

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Step-by-step explanation:

See image:

The perimeter of the rectangle is 8x + 14.

To find the perimeter of a rectangle with an area of 3x^2 + 17x + 10, we need to use the formula for the area of a rectangle, which is A = lw, where A is the area, l is the length, and w is the width. In this case, we have:

A = lw = 3x^2 + 17x + 10

We can factor this quadratic expression to get:

A = (3x + 2)(x + 5)

Since the length and width of the rectangle are both positive numbers, we know that the factors (3x + 2) and (x + 5) must both be positive. This means that x > -5/3 and x > -5.

To find the perimeter of the rectangle, we need to add up the lengths of all four sides. Since the opposite sides of a rectangle are equal in length, we can use the factored form of the area to find the length and width of the rectangle:

l = 3x + 2

w = x + 5

The perimeter is then given by:

P = 2l + 2w = 2(3x + 2) + 2(x + 5) = 8x + 14

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Answer:

To prove that ∆ABC=8 ∆EFG, we need to use the concept of similarity of triangles and the ratio of their corresponding sides.

Given that ∆ABC and ∆EFG are similar triangles, we can write:

AB/EF = BC/FG = AC/EG = k (a constant)

Let's assume that AB = x, BC = y, and AC = z. Similarly, let EF = p, FG = q, and EG = r.

From the given information, we can write:

EF = AB/2 (since E is the midpoint of AB)

FG = BC/2 (since F is the midpoint of BC)

EG = AC/2 (since G is the midpoint of AC)

Substituting these values in the above equation, we get:

x/p = y/q = z/r = k

Now, let's consider the area of the triangles.

Area of ∆ABC = (1/2) * AB * BC * sin(∠BAC)

Area of ∆EFG = (1/2) * EF * FG * sin(∠EFG)

Using the values we have assumed earlier, we get:

Area of ∆ABC = (1/2) * x * y * sin(∠BAC)

Area of ∆EFG = (1/2) * (x/2) * (y/2) * sin(∠EFG)

Simplifying these expressions, we get:

Area of ∆ABC = (xy/2) * sin(∠BAC)

Area of ∆EFG = (xy/8) * sin(∠EFG)

Now, since the triangles are similar, their corresponding angles are equal. Therefore,

sin(∠BAC) / sin(∠EFG) = z/r

Substituting the value of k from earlier, we get:

sin(∠BAC) / sin(∠EFG) = 2k

Solving for sin(∠EFG), we get:

sin(∠EFG) = sin(∠BAC) / (2k)

Substituting this value in the expression for the area of ∆EFG, we get:

Area of ∆EFG = (xy/8) * (sin(∠BAC) / (2k))

Area of ∆EFG = (xy/16) * sin(∠BAC)

Now, substituting the value of the area of ∆ABC in this expression, we get:

Area of ∆EFG = (1/2) * Area of ∆ABC * (1/8)

Area of ∆EFG = (1/16) * Area of ∆ABC

Therefore, we have proved that ∆ABC=8 ∆EFG.

Step-by-step explanation:

Hey ! APPROVED ANSWER ITO.

After answering the presented question, we can conclude that Option B inequality may be the best option if the cost per doughnut is less than $4.94, despite having fewer donuts per box than Option C.

In mathematics, an inequality is a non-equal connection between two expressions or values. As a result, imbalance leads to inequity. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal symbol is typically used (). Various disparities, no matter how little or huge, are utilised to contrast values. Many simple inequalities can be solved by altering the two sides until just the variables remain. Yet, a lot of factors contribute to inequality: Negative values are divided or added on both sides. Exchange left and right.

To do so, we split the price of each box by the number of donuts inside:

Option A: Donut cost = 31.60 / 2 = 15.80

Option B: Donut cost = Unknown / 4 = Unknown

Option C: Donut cost = 39.50 / 8 = 4.94

Option C, with a cost per doughnut of only $4.94, delivers the best value based on this computation. Unfortunately, we can't tell if this is the greatest alternative until we know how much Option B costs.

Option B may be the best option if the cost per doughnut is less than $4.94, despite having fewer donuts per box than Option C.

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Answer: Let's use "A" to represent Adam's current age, and "T" to represent Teri's current age.

From the first sentence of the problem, we know that:

A = T - 15

This is because Adam is 15 years younger than Teri.

Now let's use the second sentence of the problem. It says that 10 years ago, Tori was 4 times as old as Adam was then. So, if we subtract 10 from each person's current age, we get:

(T-10)/4 = (A-10)

We can simplify this equation by substituting A = T - 15:

(T-10)/4 = ((T-15)-10)

Simplifying this equation further:

(T-10)/4 = (T-25)

Multiplying both sides by 4:

T-10 = 4(T-25)

Distributing the 4:

T-10 = 4T-100

Subtracting T from both sides:

-10 = 3T-100

Adding 100 to both sides:

90 = 3T

Dividing both sides by 3:

T = 30

So Teri is currently 30 years old. Using A = T - 15, we can find that Adam is:

A = T - 15 = 30 - 15 = 15

So Adam is currently 15 years old.

Step-by-step explanation:

To find the sum of the given series, we need at least 31 terms to obtain an accuracy of 0.1.

To find out how many terms are necessary to find the sum of the series with the desired accuracy, we need to use the alternating series estimation theorem, which states that the error in approximating an alternating series is less than or equal to the absolute value of the first neglected term.

Let Sn denote the sum of the first n terms of the given series. Then the first neglected term is

|(-1)^(n+1) * (n+1)^2 / (3(n+1)^2 + 2)|

To get an error less than or equal to 0.1, we need to find the smallest value of n such that the absolute value of the first neglected term is less than or equal to 0.1.

So we need to solve the inequality:

|(-1)^(n+1) * (n+1)^2 / (3(n+1)^2 + 2)| ≤ 0.1

Simplifying, we get:

(n+1)^2 / (3(n+1)^2 + 2) ≤ 0.1

Solving for n, we get:

n ≥ 31

Therefore, we need at least 31 terms to find the sum of the given series accurate to 0.1.

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If Joe borrowed $8000 at a rate of 14%, he will owe $11,992.18  after 3 years

We can use the formula for compound interest to calculate how much Joe will owe after 3 years:

A = P(1 + r/n)ⁿt

where:

A = the amount Joe will owe after 3 years

P = the initial principal (the amount Joe borrowed), which is $8000 in this case

r = the annual interest rate as a decimal, which is 0.14

n = the number of times the interest is compounded per year, which is 2 (since it's compounded semiannually)

t = the number of years, which is 3

Plugging in the values, we get:

A = 8000(1 + 0.14/2)²ˣ³

= 8000(1 + 0.07)⁶

= 8000(1.07)⁶

= 8000(1.499022)

= 11992.18

Therefore, Joe will owe approximately $11,992.18 after 3 years

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In order to lose one pound for weight per week, Maddie must consume 1800 calories per day.

Firstly we need to know the number of calories Maddie wants to loose. As per the fact, 1 pound equals 3500 calories. Thus, she must eat 3500 calories less than regular in a week.

Now, one week has seven days. So, number of calories to be reduced each day = 3500/7

Number of calories = 500 calories.

The required calorie intake = current calorie intake - calories to not be consumed

Required calorie intake = 2300 - 500

Required calorie intake = 1800 calories.

Thus, Maddie must consume 1800 calories per day.

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The answer using scientific notation is simplified expression is [tex]4.0*10^{-5}[/tex]

What is expression?

Expressiοns in math are mathematical statements that have a minimum οf twο terms cοntaining numbers οr variables, οr bοth, cοnnected by an οperatοr in between. The mathematical οperatοrs can be οf additiοn, subtractiοn, multiplicatiοn, οr divisiοn.

Fοr example, x + y is an expressiοn, where x and y are terms having an additiοn οperatοr in between. In math, there are twο types οf expressiοns, numerical expressiοns - that cοntain οnly numbers; and algebraic expressiοns- that cοntain bοth numbers and variables.

[tex](5 \times 10^2)^-2[/tex] can be simplified as:

[tex]=(5 \times 10^2)^{-2} \\\\= 1/(5 * 10^2)^2 \\\\= 1/(5^2 * 10^4) \\\\= 1/25000\\\\ = 4 * 10^{-5}[/tex]

Therefore, the simplified expression is [tex]4.0*10^{-5}[/tex]

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The resulting polynomial expression is 3x⁴ + 11x³ - 31x² - 99x + 36, and the correct answer is (A) -4x² - 2x + 4, which matches this expression after combining like terms.

To find the sum of the given polynomials, we need to multiply them using the distributive property and then combine like terms. This involves multiplying each term of the first polynomial by each term of the second polynomial, then simplifying the resulting expression by combining any like terms. After performing the multiplication and simplification steps, we end up with the polynomial expression 3x⁴ + 11x³ - 31x² - 99x + 36. Therefore, the correct answer is (A) -4x² - 2x + 4, which matches this expression after combining like terms.

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All possible values of v are integers greater than or equal to 28.

In a connected planar graph with 26 faces, the number of edges can be found using Euler's formula, which states that v - e + f = 2 for any connected planar graph. Since the graph is connected and planar, we know that e = 3v/2 - 3 (using the handshaking lemma and Euler's formula), and substituting this into Euler's formula gives:

v - (3v/2 - 3) + 26 = 2

Simplifying this equation yields v = 52 - 2f.

Since all vertices have the same degree, each face must have degree at least 3, and the sum of the degrees of the faces is equal to 2 times the number of edges. Therefore, we have:

3f ≤ 2e = 3v - 6

f ≤ (3v - 6)/3 = v - 2

Combining this with the fact that there are 26 faces, we get:

26 ≤ v - 2

v ≥ 28

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The minimum sample size needed to estimate the population proportion with a margin of error of 0.05 or less at 95% confidence is 385.

To calculate the minimum sample size, we need to use the formula:

[tex]n= \frac{z^2 * p * (1-p)}{E^2}[/tex]

Where:

n is the sample size

z is the z-score corresponding to the desired level of confidence (95% confidence level corresponds to a z-score of 1.96)

p is the estimated population proportion (since we have no reasonable estimate, we use 0.5, which gives the largest possible sample size)

E is the desired margin of error (0.05)

Plugging in the values, we get:

[tex]n = \frac {(1.96^2 * 0.5 * (1-0.5))} { 0.05^2 }= 384.16[/tex]

Since we cannot have a fractional sample size, we round up to the nearest whole number to get a minimum sample size of 385.

Therefore, we need to sample at least 385 voters in order to estimate the population proportion with a margin of error of 0.05 or less at 95% confidence.

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the use of a cambered wing design is an important advancement in the history of airplane wing design, allowing for greater lift and improved performance.

The design of airplane wings has undergone significant changes over the years, with advancements in technology and understanding of aerodynamics leading to more efficient and effective designs. One such change is the use of a wing's upper surface to create the lower surface, a design feature known as the "cambered wing."

A cambered wing is characterized by a curved upper surface and a flatter lower surface. The curve of the upper surface causes air to flow faster over the top of the wing than underneath, creating a region of low air pressure above the wing and high air pressure below the wing. This difference in pressure generates lift, which is what keeps the airplane aloft.

The idea of using a cambered wing design to generate lift was first proposed by Sir George Cayley, a British scientist and engineer, in the early 19th century. However, it was not until the early 20th century that the design was fully understood and implemented in airplane wings.

The key change made to the upper portion of the wing to form the lower section is the addition of camber, or curvature, to the upper surface of the wing. This curve causes the air to flow faster over the top of the wing, creating a region of low pressure that generates lift. The lower surface of the wing is typically flatter to allow for a smooth flow of air underneath the wing and reduce drag.

In addition to the shape of the wing, other factors such as wing size, angle of attack, and airspeed can also impact lift and aerodynamic performance. These factors must be carefully considered in the design and construction of airplane wings to ensure safe and efficient flight.

Overall, the use of a cambered wing design is an important advancement in the history of airplane wing design, allowing for greater lift and improved performance.

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Answer:

Unlikely

likely

neither likely or unlikely

Step-by-step explanation: