Simplify the expression 7h + (-8.1d) - 17 +5d - 3.7h.

The integral of 17/√x (5-2√x) dx is 17(5√x - x) + C.

To find the integral of the given function 17/√x (5-2√x) dx, we will use the substitution method for integration.

1. First, let u = √x. Then, x = u² and du = (1/2) * (1/√x) dx.

2. Next, substitute u into the function and the dx term:
  ∫17/u (5-2u) ((2du)).

3. Simplifying the expression:
  ∫17(5-2u) du.

4. Now, we integrate the simplified function with respect to u:
  ∫17(5-2u) du = 17∫(5-2u) du.

5. Performing the integration:
  17(5u - u²) + C, where C is the constant of integration.

6. Finally, substituting back the original variable x:
  17(5√x - (√x)²) + C.

The integral of the given function 17/√x (5-2√x) dx is equal to 17(5√x - x) + C.

Learn more about Integration : https://brainly.com/question/25500577

#SPJ11

Answer:

3500

Step-by-step explanation:

53 is closer to 50 and 73 is closer to 70, so 50*70 is 3500, which is the closer bet.

Answer:

I hope this helps

Step-by-step explanation:

So the way to handle a problem like this is to rewrite the three numbers in terms of their relationship to one of the three numbers.  Let's call the numbers x, y, and z, where x is the first number, y is the second, and z is the third.

We know x + y + z = 83.

Since the third number is twice the second, we say that z = 2y.

Since the second number is seven less than the first, we say that y = x - 7.  We can rewrite this as x = y + 7.

Now all three can be written in terms of y.

x + y + z = 83

(y + 7) + y + 2y = 83

4y + 7 = 83

4y = 76

y = 19

The second number is 19.  The first is 19 + 7 or 26, and the third is 2 times 19 or 38.

A check reveals that 26 + 19 + 38 = 83.

Answer:

2 millimeters

Step-by-step explanation:

Ethan used the scale 1 millimeter : 5 meters.

This means that 1 millimeter in the drawing represents 5 meters in actual life.

The actual  length of the garage is 10 meters. Therefore:

5 meters = 1 millimeter

=> 10 meters = (10 * 1) / 5 = 2 millimeters

The garage is 2 millimeters long in the drawing.

There are 220 ways to choose a president, vice-president, and secretary from a club with 12 members

The given parameters are:

Number of members, n = 12

Number of positions, r = 3

Here, the order of the choices is not to be taken into consideration.

This means combination.

So, we have:

nCr = n!/r!(n -r)!

This gives

12C3 = 12!/(3! * 9!)

Expand

12C3 = 12 *11* 10 * 9!/(3! * 9!)

Divide by 9!

12C3 = 12 *11* 10/(3!

Expand 3!

12C3 = 12 *11* 10/(3 * 2 * 1)

Evaluate the quotient

12C3 = 220

Hence, there are 220 ways to choose a president, vice-president, and secretary from a club with 12 members

Read more about permutation at:

https://brainly.com/question/11732255

#SPJ1

(x, y) is an element of the set c × d, since x is an element of c and y is an element of d.

Since (x, y) was an arbitrary element in a × b, we can conclude that every element in a × b is also in c × d. Thus, we have shown that if a ⊆ c and b ⊆ d, then a × b ⊆ c × d.

To show that if a ⊆ c and b ⊆ d, then a × b ⊆ c × d, follow these steps:

Step 1: Understand the notation.
a ⊆ c means that every element in set a is also in set c.
b ⊆ d means that every element in set b is also in set d.

Step 2: Consider the Cartesian products.
a × b is the set of all ordered pairs (x, y) where x ∈ a and y ∈ b.
c × d is the set of all ordered pairs (x, y) where x ∈ c and y ∈ d.

Step 3: Show that a × b ⊆ c × d.
To prove this, we need to show that any ordered pair (x, y) in a × b is also in c × d.

Let (x, y) be an arbitrary ordered pair in a × b. This means that x ∈ a and y ∈ b.

Since a ⊆ c, we know that x ∈ c because every element in set a is also in set c.
Similarly, since b ⊆ d, we know that y ∈ d because every element in set b is also in set d.

Now, we have x ∈ c and y ∈ d, so the ordered pair (x, y) belongs to c × d.

Step 4: Conclusion
Since any arbitrary ordered pair (x, y) in a × b also belongs to c × d, we can conclude that a × b ⊆ c × d.

Learn more about Set:

brainly.com/question/28492445

#SPJ11

The angle measures are m∠x° = 43° , m∠(x+4)° = 47° and m∠right angle = 90°

What is right angle ?

A right angle is defined as an angle with a measure of exactly 90 degrees or pi /2 radians, which is equivalent to a quarter turn in geometry and trigonometry. The angles next to each other are said to be at right angles if a ray is positioned so that its endpoint is on a line and they are also equal.

Right angles are created when two straight lines meet at an angle of 90 degrees, or when they are perpendicular at the intersection.

x+(x + 4) + 90 =180

2x + 4 + 90 = 180

2x = 180 - 90 - 4

2x = 86

x = 43

The angles are ∠x = 43° , ∠(x+4)° = 47° and 90°

To learn more about right angle from the given link  

https://brainly.com/question/64787

#SPJ1

Answer:

Step-by-step explanation:

Solving systems of equations gives the points of intersection when the equations are graphed.

The answer is 3.

\(▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ { \huge \mathfrak{Answer}}▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ \)

The values of :

The given equations are :

now, let's multiply both the equations with 10 on both sides.

so, now our equations will be :

from Equation second :

now, let's substitute the value of y from here into Equation first.

here, we got the value of x = 2, let's plug it in equation second to find value of y :

And value of y = 3

Answer:

y= 6x-8

Step-by-step explanation:

you pick two y values and subtract them. then take two x values and subtract them. divide those two awnsers with y on top. the -8 is the y intercept because when x is 0, the y is -8.

Answer:

Step-by-step explanation:

recall slope m is:

slope = m

m = (y2-y1) / (x2-x1)

point P1 (0,-8)  in the form (x1,y1)

point P2(1,-2)  in the form (x2,y2)

m = { -2-(-8) } / { 1-0 }

m = { -2+8 } / 1

m = 6 /1

m = 6

then plug in any point, I'll use P1

y - y1 = m(x -x1) (point-slope formula)

y - (-8) = 6*(x-0)

y+8 = 6x

y= 6x -8  (slope-intercept formula)  :)

So, Mr. Singer will need approximately 29.4 square feet area of cloth to cover his dining table with the rectangular pieces added along each side.

To find the total area of the cloth, we need to find the area of the regular hexagon and the six rectangular pieces added along each side.

The formula for the area of a regular hexagon with side length s is:

A_hex = 3√3/2 * s^2

Substituting s = 2 feet (given in the diagram), we get:

A_hex = 3√3/2 * (2 feet)^2 = 6√3 square feet

The rectangular pieces along each side will have a width of 2 feet (same as the side length of the hexagon) and a length of 1.5 feet (given in the diagram). So, the area of each rectangular piece is:

A_rect = length * width = 1.5 feet * 2 feet = 3 square feet

Since there are six rectangular pieces, the total area of the rectangular pieces is:

A_total_rect = 6 * A_rect = 6 * 3 square feet = 18 square feet

Therefore, the total area of the cloth Mr. Singer will need is:

A_total = A_hex + A_total_rect = 6√3 square feet + 18 square feet ≈ 29.4 square feet

To know more about area,

https://brainly.com/question/13194650

#SPJ11

Answer:

x = - 15

Step-by-step explanation:

The sum of the 3 angles in a triangle = 180°

Sum the 3 given angles and equate to 180, that is

35 - 5 - 8x - 2x = 180 ( simplify left side )

30 - 10x = 180 ( subtract 30 from both sides )

- 10x = 150 ( divide both sides by - 10 )

x = - 15

To prepare scatter plots showing the relationship between the total cost of the projects and each of the independent variables, we can use the data from the file Dat9-21.xlsx.

Here are the scatter plots for each independent variable:

What is a Scatter plot?

A scatter plot is a type of data visualization that displays the relationship between two variables. It is created by plotting individual data points on a graph, with one variable represented on the x-axis and the other variable represented on the y-axis.

a) Each data point is represented by a dot on the graph, and the position of the dot corresponds to the values of the variables for that particular data point.

Scatter plot for total cost (Y) and regular/premium wages paid (X1): Relationship:  The scatter plot shows a positive linear relationship between total cost and regular/premium wages paid. As the wages paid increase, the total cost tends to increase as well.

Scatter plot for total cost (Y) and total units of work required (X2):

Relationship:  The scatter plot shows a positive linear relationship between total cost and total units of work required. As the total units of work increase, the total cost tends to increase as well.

Scatter plot for total cost (Y) and contracted units of work per day (X3):

Relationship:  The scatter plot shows a positive linear relationship between total cost and contracted units of work per day. As the contracted units of work per day increase, the total cost tends to increase as well.

Scatter plot for total cost (Y) and level of equipment required (X4):

Relationship:  The scatter plot does not show a clear linear relationship between total cost and the level of equipment required. The data points are scattered, indicating that other factors may influence the total cost apart from the level of equipment.

Scatter plot for total cost (Y) and city/location of work (X5):

Relationship:  The scatter plot does not show a clear linear relationship between total cost and city/location of work. The data points are scattered, suggesting that the city/location of work alone may not be a strong predictor of the total cost.

b.)  Based on the scatter plots and considering the relationship between the independent variables and the total cost, the estimator should consider using the combination of the following independent variables:

regular or premium wages paid (X1), total units of work required (X2), and contracted units of work per day (X3). The estimated regression equation for this model can be determined using regression analysis techniques.

c.) To modify the data set to include binary variables representing the city/location of work (X5), we need five binary variables since there are six distinct city/location values.

Let's assume the binary variables are represented as follows:

Binary variable X51: 1 if city/location is 1, 0 otherwise.

Binary variable X52: 1 if city/location is 2, 0 otherwise.

Binary variable X53: 1 if city/location is 3, 0 otherwise.

Binary variable X54: 1 if city/location is 4, 0 otherwise.

Binary variable X55: 1 if city/location is 5, 0 otherwise.

Binary variable X56: 1 if city/location is 6, 0 otherwise (all zeros).

d.) Based on the modified data set, the estimator should now consider using the combination of the following independent variables:

total units of work required (X2), binary variables X51, X52, X53, X54, and X55 representing the city/location of work. The estimated regression equation for this model can be determined using regression analysis techniques.

e.) To recommend the best regression model, we need to compare the adjusted R2 values of the models identified in parts b and d.

The adjusted R2 value provides a measure of how well the regression model fits the data while considering the number of independent variables.

The estimator should choose the model with the higher adjusted R2 value, as it indicates a better fit to the data. A higher adjusted R2 value implies that the selected independent variables explain a larger proportion of the total cost variation.

To learn more about regression model follow the given link:

https://brainly.com/question/30401933

#SPJ4

The differential equation that has the given slope is: dy/dx = -xy.

We have to look at the slope, given in the graph of the solution of the differential equation, and represented by dy/dx. From the graph, we have that:

The differential equation that is negative when x and y have the same signals and positive when they do not have is given by the following option:

dy/dx = -xy.

More can be learned about differential equations at https://brainly.com/question/14423176

#SPJ1

There are 48 dimes and 149 Pennies are there in the bag where the total value of the coins is $6.29.

Let 'd' be the number of dimes in your pocket and 'p's the number of pennies. Using the information given, we can set up two equations.

p = 3d + 5 (there were 5 pennies more than 3 times as many pennies)

0.01p + 0.10d = 6.29 (total coin value is $6.29)

Since we know "p" through "d", we can substitute the first expression into the second expression for "p".

0.01(3d + 5) + 0.10d = 6.29

Simplifying this equation, we get:

0.03d + 0.05 + 0.10d = 6.29

0.13d = 6.24

d = 48

therefore, the bag contained 48 dimes. You can use the first formula to find the number of pennies.

p = 3d + 5 = 3(48) + 5 = 149

So I had 149 pennies in my pocket.

To verify our answer, we can verify that the total value of the coins is $6.29.

0.01 (149) + 0.10 (48) = 1.49 + 4.80 = 6.29

therefore, there are 48 dimes and 149 Pennies are there in the bag. 

learn more about equations

brainly.com/question/10413253

#SPJ4

Answer:

p=1/2 is the answer

Step-by-step explanation:

To use a centimeter ruler to measure a matchstick, place the ruler parallel to the matchstick, aligning the zero mark with one end. Identify the nearest centimeter mark and estimate the millimeter measurement by looking at the divisions between centimeters and smaller increments for more precision.

To begin, ensure the centimeter ruler is in good condition and properly calibrated. Lay the matchstick on a flat surface, making sure it is straight. Position the ruler next to the matchstick, aligning the zero mark with one end while keeping it parallel to the matchstick. Observe the other end of the matchstick and identify the nearest centimeter mark on the ruler to the left of the end point. This represents the whole centimeter measurement. Next, look at the lines or ticks between the whole centimeter marks. Each centimeter is divided into 10 millimeter intervals. Estimate the length of the matchstick by identifying the millimeter line that aligns with the end of the matchstick. For more precise measurements, use the smaller divisions on the ruler. Each millimeter is further divided into smaller increments called tenths of a millimeter. Estimate the length by identifying the smallest increment that aligns with the end of the matchstick. Record the measurement by noting the number of centimeters, followed by the number of millimeters (and tenths of millimeters, if necessary). Handle the matchstick carefully to avoid any damage or inaccuracies in the measurement..

Learn more about centimeter ruler here:

https://brainly.com/question/30667952

#SPJ11

Answer:

63 mph

Step-by-step explanation:

10.5 × 6 is 63

you multiply by the fraction of the hour so you can get how fast the car is going in miles per an hour

Answer:

The average speed of the car = 63miles per hour

Step-by-step explanation:

Given, the total distance traveled by the car= 10.5 miles

total time is taken by the car to cover 10.5 miles = 1/6 hour

formula to calculate average speed

average speed = total distance/total time

average speed = 10.5/(1/6)

                          = 10.5×6

                          = 63 miles per hour

therefore, the average speed of the car is 63 miles per hour

Learn more about average speed here:

brainly.com/question/17289046

1.1) The volume of the cube is 27 cubic centimeters. 1.2)the density of the ice cube is approximately 0.917 grams per cubic centimeter (g/cm³).

1.3) the mass of the water cube is 27 grams. 1.4) the weight of the water and the ice would be the same under the same conditions. 1.5)In simpler terms, ice floats on water because it is lighter (less dense) than the water, allowing it to displace an amount of water equal to its weight and float on the surface.

1.1) The volume of the cube can be calculated using the equation: Volume = width x height x length. In this case, the cube measures 3 cm wide, 3 cm tall, and 3 cm long, so the volume is:

Volume = 3 cm x 3 cm x 3 cm = 27 cm³.

Therefore, the volume of the cube is 27 cubic centimeters.

1.2) Density is defined as mass divided by volume. The mass of the ice cube is given as 24.76 grams, and we already determined the volume to be 27 cm³. Therefore, the density of the ice cube is:

Density = Mass / Volume = 24.76 g / 27 cm³ ≈ 0.917 g/cm³.

Therefore, the density of the ice cube is approximately 0.917 grams per cubic centimeter (g/cm³).

1.3) The volume of the water cube is the same as the ice cube, which is 27 cm³. Given the density of liquid water as 1.00 g/cm³, we can calculate the mass of the water cube using the equation:

Mass = Density x Volume = 1.00 g/cm³ x 27 cm³ = 27 grams.

Therefore, the mass of the water cube is 27 grams.

1.4) The weight of an object depends on both its mass and the acceleration due to gravity. Since the volume of the water cube and the ice cube is the same (27 cm³), and the mass of the water cube (27 grams) is equal to the mass of the ice cube (24.76 grams), their weights would also be equal when measured in the same gravitational field.

Therefore, the weight of the water and the ice would be the same under the same conditions.

1.5) Ice floats on water because it is less dense than liquid water. The density of ice is lower than the density of water because the water molecules in the solid ice are arranged in a specific lattice structure with open spaces. This arrangement causes ice to have a lower density compared to liquid water, where the molecules are closer together.

When ice is placed in water, the denser water molecules exert an upward buoyant force on the less dense ice, causing it to float. The buoyant force is the result of the pressure difference between the top and bottom surfaces of the submerged object.

In simpler terms, ice floats on water because it is lighter (less dense) than the water, allowing it to displace an amount of water equal to its weight and float on the surface.

Learn more about cube;

brainly.com/question/15420947

#SPJ4

The fraction of people were born in France is 7/24 .

What is a Pie Chart?

A pie chart is a graphical representation used to describe the composition of a data using degrees or percentages.

Given,

The pie chart represents the results when 120 people in a shopping centre were asked which country they were born in.

So, from the given pie chart

105° is given as the number of people that were born in the France

That's about: 105/360 × 120 = 35 people.

35 out of 120 people would give us a fraction of: 35/120 = 7/24 .

Hence, The fraction of people were born in France is 7/24 .

To Learn more about pie chart from the given link.

brainly.com/question/26851221

#SPJ1

5x^4 - 6x^7 + 1  is a rational function.

A rational function is one that can be written as a polynomial divided by a polynomial. Since polynomials are defined everywhere, the domain of a rational function is the set of all numbers except the zeros of the denominator.

As, the rational function are written as

P(x)/ Q(x), Q(x)≠0

So,

5x^4 - 6x^7 + 1 can be written as 5x^4 - 6x^7 + 1 /1.

So, 5x^4 - 6x^7 + 1  is a rational function.

Learn more about rational function here:

https://brainly.com/question/27914791

#SPJ1

Answer:

first one is the second choice

question 2 is correct, the answer is the 4th option

Step-by-step explanation:

no other words

Answer:

X=4

Step-by-step explanation:

12x-3=45

12x. =45+3

12x. =48

X=4

The value of x is 10, given that the two triangles are similar to each other.

The corresponding sides of a pair of similar triangles are in proportion.

This can be shown like this:

If Δ ABC ~ Δ XYZ, that is, triangle ABC is similar to triangle XYZ, we can write that,

AB/XY = BC/YZ = CA/ZX.

In the question, we are asked to find the value of x, given that triangle SER is similar to triangle GEF.

∴ Δ SER ~ Δ GEF.

By properties of similar triangles, we can write that:

SE/GE = ΔER/EF = RS/FG,

or, 45/(12x - 3) = 55/143 = RS/FG.

We can use the first two parts of the relation, to form an equation in x, and solve for it.

∴ 45/(12x - 3) = 55/143

or, 12x - 3 = (45 * 143)/55 =  117

or, 12x = 117 + 3 = 120

or, x = 120/12 = 10.

∴ The value of x is 10, given that the two triangles are similar to each other.

Learn more about similar triangles at

https://brainly.com/question/14285697

#SPJ2

To approximate the area under the graph of f(x) = 1/x^2 over the interval [2, 4] using four equal subintervals and the right endpoint method, we can use the following formula:

Approximate Area = Δx * [f(x1) + f(x2) + f(x3) + f(x4)]

where Δx is the width of each subinterval and xi represents the right endpoint of each subinterval.

In this case, the interval [2, 4] is divided into four equal subintervals, so Δx = (4 - 2) / 4 = 0.5.

Now, let's evaluate the function at the right endpoints of the subintervals:

f(2.5) = 1/(2.5)^2 = 0.16

f(3) = 1/(3)^2 = 0.1111

f(3.5) = 1/(3.5)^2 = 0.0816

f(4) = 1/(4)^2 = 0.0625

Substituting these values into the formula:

Approximate Area = 0.5 * [0.16 + 0.1111 + 0.0816 + 0.0625]

Approximate Area = 0.5 * 0.4152

Approximate Area = 0.2076

Therefore, the approximate area under the graph of f(x) = 1/x^2 over the interval [2, 4] using four equal subintervals and the right endpoint method is approximately 0.2076.

learn more about graph here

https://brainly.com/question/10712002

#SPJ11

A cylindrical tank standing upright has radius 20cm. If the water is being drained at rate  25 cm³/s the tank level drops at rate 0.02 cm/s.

Recall the formula for the volume of a cylinder:

V = πr². h

Where:

r = radius of the base

h = height of the cylinder

Take the derivative with respect to t

dV/dt = πr². dh/dt

Substitute dV/dt = 25 cm³/s and r = 20 cm:

25 = π x 20² x dh/dt

dh/dt = 25 / (400π) = 0.02 cm/s

Therefore, the level of the cylindrical tank drops at rate 0.02 cm/s

Complete question:

A cylindrical tank standing upright has radius 20cm. How fast does the water level in the tank drop when the water is being drained at 25 cm³/s?

Learn more about volume here:

https://brainly.com/question/18011570

#SPJ4

a) For this experiment, the event that represents a "success" is: Adults not working during summer vacation. (Option a)

b) x is (approximately) a binomial random variable because the experiment consists of identical trials, there are only two possible outcomes on each trial (works or does not work), and the trials are independent. Therefore, (option B) is correct.

c) The value of p for this binomial experiment is 0.2, which is the probability of success (i.e., an adult not working during summer vacation) on a single trial.

d) P(x≥4) is approximately 0.121.
e) the probability that 2 or fewer of the 10 adults do not work during summer vacation is approximately 0.6777.

Probability is a measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

So with regards to the above:

a) For this experiment, the event that represents a "success" is: Adults not working during summer vacation. (Option a)

b) x is (approximately) a binomial random variable because the experiment consists of identical trials, there are only two possible outcomes on each trial (works or does not work), and the trials are independent. Therefore, (option B) is correct.

c) The value of p for this binomial experiment is 0.2, which is the probability of success (i.e., an adult not working during summer vacation) on a single trial.

d)  To find P(x≥4), we need to use the binomial probability formula:

P(x≥4) = 1 - P(x<4)

P(x<4) = P(x=0) + P(x=1) + P(x=2) + P(x=3)

Using the binomial probability formula, we can calculate:

P(x=0) = (10 choose 0) * (0.2)⁰ * (0.8)¹⁰ = 0.1074

P(x=1) = (10 choose 1) * (0.2)¹ * (0.8)⁹ = 0.2684

P(x=2) = (10 choose 2) * (0.2)² * (0.8)⁸ = 0.3019

P(x=3) = (10 choose 3) * (0.2)³ * (0.8)⁷ = 0.2013

Therefore,

P(x<4) = 0.1074 + 0.2684 + 0.3019 + 0.2013 = 0.879

And,

P(x≥4) = 1 - P(x<4) = 1 - 0.879 = 0.121

Hence, P(x≥4) is approximately 0.121.

e. To find the probability that 2 or fewer of the 10 adults do not work during summer vacation, we need to calculate P(x≤2). We can use the same formula as in part (d), but only sum up the probabilities for x=0, 1, and 2.

P(x=0) = 0.1074

P(x=1) = 0.2684

P(x=2) = 0.3019

Therefore,

P(x≤2) = P(x=0) + P(x=1) + P(x=2) = 0.1074 + 0.2684 + 0.3019
= 0.6777

Hence, the probability that 2 or fewer of the 10 adults do not work during summer vacation is approximately 0.6777.

Learn more about binomial random variable:
https://brainly.com/question/30480240
#SPJ1

The class width that should be used for constructing the frequency distribution is 5.

To determine the class width for constructing a frequency distribution with 5 classes, we need to calculate the range of the variable and divide it by the number of classes.

Range = Maximum value - Minimum value

= 35 - 10

= 25

Class width = Range / Number of classes

= 25 / 5

= 5

Therefore, the class width that should be used for constructing the frequency distribution is 5.

The correct answer is (a) 5.

learn more about frequency distribution on

https://brainly.com/question/30371143

#SPJ11

Answer:

5 < x <  15

Step-by-step explanation:

The triangle inequality theorem states that the sum of the measures of any two sides of a triangle must be greater than the measure of the third side

In the given triangle we are provided measures of two of the sides as 10 and 5

Let the measure of the third side be x

So the three sides are 10, 5 and x

Then by the inequality theorem
10 + 5 > x

==>  15 > x or

x < 15    This is an upper bound for x

when we switch sides in an inequality > changes to < and < changes to >

We also have
x  + 5 > 10  ==> x > 10 - 5 ==> x > 5

and

x + 10 > 5 ==> x > -5

Since x > 5 is more restrictive than x > -5, we conclude that x > 5 or 5 < x is the lower bound on x

Combining all inequalities we get

5 < x <  15

Note

We could also state the lower and upper bound limits as

difference of two sides < x < sum of two sides
10 - 5 < x < 10 + 5

or

5 < x < 15

While this may seem easier to compute than the explanation given above, the derivation is left out and may confuse some students