The system of linear equations represents the situation is;
x + y = 125
x + y = 1255x + 8y = 775
Simultaneous equationSimultaneous equation is an equation in two unknown values are being solved for at the same time.
let
number of quick washes = xnumber of premium washes = yx + y = 125
5x + 8y = 775
From equation (1)
x = 125 - y
5x + 8y = 775
5(125 - y) + 8y = 775
625 - 5y + 8y = 775
- 5y + 8y = 775 - 625
3y = 150
y = 150/3
y = 50
Substitute y = 50 intox + y = 125
x + 50 = 125
x = 125 - 50
x = 75
Therefore, the number of quick washes and premium washes Monica’s school band had is 75 and 50 respectively.
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Answer:
A. 5x + 8y = 775 and x + y =125
Step-by-step explanation:
help!! what do i do?!?
What is the solution to this inequality?
7x+18>−3
x>−3
, , x greater than negative 3, ,
x<−3
, , , , x less than negative 3, , , ,
x>3
, , x greater than 3, ,
x<3
Answer:
x>-3
Step-by-step explanation:
7x+18>-3
transpose 18 to the other side to group the like terms
7x>-3-18
7x>-21
You divide bother sides by a 7 to get the value of x
7x/7>-21/7
therefore x>-3
The motion of an object was recorded by measuring the velocity of the object in meters per second at various times, as shown in the table. The scatterplot below was graphed from the data in the table, where t represents the time, and v represents the velocity. Based on the scatterplot, which equation BEST represents the relationship between velocity and time?
Based on the scatterplot, the equation that BEST represents the relationship between velocity and time is y = 2.2x + 6
How to determine the line of best fit?The line of best fit is the line that best represents the relationship between 2 variables. The equation of a line in slope-intercept form is expressed as:
y = mx + b
where
m is the slope
b is the intercept
Using the coordinate points on the plane (0, 6) and (10, 28)
Determine the slope
Slope = 28-6/10-0
Slope = 22/10
Slope = 2.2
Determine the y-intercept
6 = 2.2(0) + b
b = 6
Determine the equation of aline of best fit.
y = 2.2x + 6
Hence based on the scatterplot, the equation that BEST represents the relationship between velocity and time is y = 2.2x + 6
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Can someone tell me what this symbol means? Ty!!!
Answer:
Angle symbol
Angle ACB
Angle ACE
Angle DCE
Anja collected data about the number of dogs 9th-grade students own. She created this histogram to represent the data and determined that it is skewed right.
A graph shows number of dogs labeled 0 to 2 on the horizontal axis and frequency on the vertical axis. 0 dogs at frequency 20. 1 dog at frequency 15. 2 dogs at frequency 12.
Which statement is true about Anja’s claim?
Anja is correct that the histogram is skewed to the right
According to the statement
we have given that the anja collected the data and created the histogram and after analyzing we have to tell that her claim is correct or not.
So, she collected the data:
A graph shows number of dogs labeled 0 to 2 on the horizontal axis and frequency on the vertical axis. 0 dogs at frequency 20. 1 dog at frequency 15. 2 dogs at frequency 12.
we know that the
Histogram is is an approximate representation of the distribution of numerical data.
and after making the histogram we see that
The histogram is skewed to the right because there is less data on the right side.
So, Anja is correct that the histogram is skewed to the right.
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Answer:
D: She is correct; the histogram is skewed to the right because there is less data on the right side.
Which expression converts 100 inches per minute to feet per minute? 100 inches/ 1 minute x 60 minutes/ 1 hour
Option ( C ) is correct for this expression . 100 inches Over 1 minute times × 1 foot Over 12 inches.
What is a basic expression?
Expressions are basically the building blocks of Statements, in that every BASIC statement is made up of keywords (like GOTO, TO, STEP) and expressions. So expressions include not just the standard arithmetic and boolean expressions (like 1 + 2), but also values (scalar variables or arrays), functions, and constants.Given that the expression 100 inches.
We need to convert 100 inches per minute to feet per minute.
Since, we know that 1ft = 12 inch
Then,
1 in = 1/12 ft
Now, we shall convert 100 inches per minute to feet per minute.
To convert in/min to ft/min, let us multiply by 1/12
Thus, we have,
[tex]\frac{100 in}{min} * \frac{1 ft }{12 in}[/tex]
Therefore, option ( c ) is correct for this expression .
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The complete question is -
Which expression converts 100 inches per minute to feet per minute?
A) Start Fraction #1 100 inches Over 1 minute End Fraction × Start Fraction 60 minutes Over 1 hour End Fraction
B) Start Fraction #2 100 inches Over 1 minute End Fraction × Start Fraction 1 hour Over 60 minutes End Fraction
C) 100 inches Over 1 minute times × Start Fraction 1 foot Over 12 inches End Fraction
D) 100 inches Over 1 minute times × Start Fraction 12 inches Over 1 foot End Fraction
Which unit rate is the lowest price per ounce
Choice a: 10 ounces of chocolate chips for $1.66
Choice b: 20 ounces of chocolate chips for $3.22
1: choice a
2: choice b
3:unit rates are equal
4:Cannot be determined
Answer:
2: choice b
Step-by-step explanation:
a: $1.66 / 10 oz = $0.166/oz
b: $3.22 / 20 oz = $0.161/oz
Answer: 2: choice b
magan wants to ride his bicycle 30.4 miles this week. he has already ridden 4 miles if he rides for 4 more days what is the average number of miles he would have to ride each day to meet his goal?
The average number of miles he will ride each day to meet his goal is 6.6 miles
How to find the average number of miles she will ride?He wants to ride his bicycle 30.4 miles this week.
He has already ridden 4 miles.
He rides 4 more days.
Therefore, the average number of miles he would have to ride each day to meet his goal can be calculated as follows:
let
x = number of miles driven
Hence, the equation can be represented as follows;
4x + 4 = 30.4
Therefore,
4x + 4 = 30.4
4x + 4 - 4 = 30.4 - 4
4x = 26.4
x = 26.4 / 4
x = 6.6
Therefore, the average number of miles he will ride each day to meet his goal is 6.6 miles
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A company reduced the prices of its goals by 13% during a promotional sale. A product was sold for 435.00 during the promotional sales. What was the price before the promotional sales?
The price of the product before promotional sales was 500,000.
How to find out what the previous price was?To find what the previous price of the product was, we must take into account that we know that 87% of the value is equal to 435,000. So to find out how much the 13% (discount) that the product had is equivalent to, we do the following operation:
435,000 × 100 ÷ 87 = 500,000
So we know that the 13% discount was 65,000
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Given a point A(-2,5), find the coordinates of a point B so that the line segment AB has each slope.
a) 2/3
b) -2/3
c) 4
d) -3
e) 0
f) undefined
Step-by-step explanation:
I understand this that we need to find a point B for each of a) to f).
remember, the slope is the (y coordinate difference / x coordinate difference) when going from one point on the line to the other.
a)
the slope 2/3 means that when going to B x has to increase by 3, and y has to increase by 2.
so, B = (-2 + 3, 5 + 2) = (1, 7)
b)
-2/3 means that either x or y have to decrease and the other to increase.
e.g. x increases by 3, y decreases by 2.
so, B = (-2 + 3, 5 - 2) = (1, 3)
c)
4 stands for 4/1.
so, x increases by 1, y increases by 4.
B = (-2 + 1, 5 + 4) = (-1, 9)
d)
similar to c) -3 stands for -3/1.
and similar to b) either x or y has to decrease and the other to increase.
e.g. x increases by 1, y decreases by 3.
B = (-2 + 1, 5 - 3) = (-1, 2)
e)
that means the y difference is 0. B has the same y coordinate.
e.g. B = (3, 5)
x can be any value.
the line is a horizontal line going through y = 5 or (0, 5) on the y-axis.
f)
that means the x difference is 0. B has the same x coordinate.
e.g. B = (-2, 9)
y can be any value.
the line is a vertical line going through x = -2 or (-2, 0) on the x-axis.
The table below shows the birth rate (per 1000) per year in the United States according to data from the National Center for Health Statistics. Let x represent the number of years since 2000 with x = 0 representing the year 2000. Let y represent the birth rate per 1000 population. Write the slope-intercept form of the equation for the line of fit using the points representing 2001 and 2010. Round to the nearest hundredth.
The regression linear function, in slope-intercept form, is given by:
y = -0.07x + 14.5
How to find the equation of linear regression using a calculator?To find the equation, we need to insert the points (x,y) in the calculator.
For this problem, the points are given as follows:
(0, 14.7), (1, 14.5), (2, 14.4), (3, 14.1), (4, 14), (5, 14.1), (6, 14.1), (7, 14.2), (8, 14), (9, 13.8), (10, 14).
Inserting these points into a calculator, the regression linear function, in slope-intercept form, is given by:
y = -0.07x + 14.5
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Answer:
y = -0.07x + 14.5
Step-by-step explanation:
The population density of New Jersey is about 1,195 people per square mile. Texas has a land area of 261,797 square miles. If Texas had the same population density as New Jersey, how many people would be living in that state?
261,797
8,864,926
26,059,203
312,847,415
Answer:
312,847,415
Step-by-step explanation:
1. Define the given values:
population density of new jersey = 1,195 people / square mile.Texas land area = 261,797 square miles.2. Now interpret what other information the question gives:
the question states: "Texas has the same population density as New Jersey". therefore, we also know:population density of Texas = 1,195 people / square mile3. upscale the ratio:
1,195 people : 1 square mile:now we have all the information to complete the equation.if there are 1,195 people for every 1 square mile, we just have to multiply the equation by the total square miles in texas (261,797 square miles). This will give us the:total population : total square miles in Texas1,195 x 261,797 : 1 x 261,797 = 312,847,415 (total population) : 261,797 square miles in Texastherefore, the total population in Texas = 312,847,415
Hope this helps :)
PLEASE HELP and EXPLAIN I will mark the brainliest if two people type the answer I will choose the best.
Answer:
C
Step-by-step explanation:
Let's review some sets of numbers.
Natural numbers: 1, 2, 3, 4, ...
The natural numbers are the counting numbers. You normally start counting with 1, not 0, and you only count whole numbers.
Whole numbers: 0, 1, 2, 3, 4, 5, ...
The whole numbers are the counting numbers with the addition of zero. There are no negative numbers in the whole numbers.
Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
The integers are the natural numbers, plus zero, plus the negatives of the natural numbers. All numbers are whole with no decimal part.
Rational numbers
The rational numbers are all numbers that can be written as a fraction of integers. For example, 1 can be written as 3/3, so 1 is a rational number. All natural numbers, whole numbers, and integers are rational numbers. Other rational numbers are numbers that are not whole, such as 2.5, 7/8, 1/4, etc. These numbers can be written as fractions of integers. Rational numbers written as decimal numbers will either be an integer, a terminating decimal, or a non-terminating, repeating decimal.
Up to here, each new set includes the previous set.
Now things change. If a number is rational, it is not irrational.
If a number is irrational, it is not rational.
Irrational numbers
Irrational numbers are always non-terminating, non-repeating decimals. They cannot be written as fractions of integers. Examples: √2, π
Now let's look at this problem.
We have a negative decimal number. The bar above the 12 means the digits 12 are repeating. This number can also be written as -0.121212...
Since our number is a non-terminating, repeating decimal, it is a rational number.
Answer: C
Find the volume. 74yd 54yd 84yd 103yd
The volume of the figure in the attachment is 84 cubic yards
What is the volume of a figure?The volume of a figure or a three-dimensional shape is the amount of space inside the figure or the three-dimensional shape
How to determine the total volume?The complete question is added as an attachment.
From the attached figure, we can see that the three-dimensional shape is a frustum
The volume is then calculated using the following formula
V = 1/2 * (Sum of the parallel bases) * height * side length
From the figure, we have:
Parallel bases = 2 yards and 6 yards
Height = 3.5 yards
Side length = 6 yards
Substitute the known values in the above equation
V = 1/2 * (2 + 6) * 3.5 * 6
Evaluate the product in the above equation
V = 1/2 * (2 + 6) * 21
Evaluate the sum in the above equation
V = 1/2 * 8 * 21
Evaluate the product in the above equation
V = 1/2 * 168
Evaluate the product in the above equation
V = 84
Hence, the volume of the figure in the attachment is 84 cubic yards
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The population of a bacteria culture after t hours is given by P = 400 - 8t2. What is the rate of change at 6 hours?
1. 0
2. -56
3. -96
4. -48
The rate of change at 6 hours is -96
How to determine the rate of change?The function is given as:
P = 400 - 8t^2
Differentiate the function
P' = -16t
Set t = 6
P' = -16 * 6
Evaluate
P' = -96
Hence, the rate of change at 6 hours is -96
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The following are the ages of 13 history teachers in a school district. 24, 27, 29, 29, 35, 39, 43, 45, 46, 49, 51, 51, 56 Notice that the ages are ordered from least to greatest. Give the five-number summary and the interquartile range for the data set. Five-number summary
Minimum:
Lower quartile:
Median:
Upper quartile:
Maximum: Interquartile range:
The Five number summary for the given data:
Minimum: 24
Lower quartile: Q1 = 29
Median: 43
Upper quartile: Q3 = 50
Maximum: 56
Interquartile range: IQR = 21
What is the interquartile range?The interquartile range is calculated by
IQR = Q3 - Q1
Where Q3 - upper quartile and Q1 - lower quartile
Q3 = 3/4(n + 1) th term
Q1 = 1/4(n + 1) th term
Calculation:The given data points are
24, 27, 29, 29, 35, 39, 43, 45, 46, 49, 51, 51, 56
where n = 13
Maximum data value = 56
Minimum data value = 24
Calculating Median:
Median = (n+1)/2 th term
⇒ Median = (13 + 1)/2 = 7th term
∴ Median = 43
Calculating the quartiles:
Upper quartile Q3 = 3/4(n + 1)th term
⇒ Q3 = 3/4(13 + 1) = 10.5
⇒ Q3 = 10th term + 1/2(11th term - 10th term)
⇒ Q3 = 49 + 1/2(51 - 49)
⇒ Q3 = 49 + 1
∴ Q = 50
Lower quartile Q1 = 1/4(n + 1)th term
⇒ Q1 = 1/4(13 + 1) = 3.5
⇒ Q1 = 3rd term + 1/2(4th term - 3rd term)
⇒ Q1 = 29 + 1/2(29 - 29)
∴ Q1 = 29
Calculating the IQR:
IQR = Q3 - Q1
= 50 - 29
= 21
Thus, the interquartile range is 21.
Therefore, the five-number summary for the given data:
Minimum: 24
Lower quartile: Q1 = 29
Median: 43
Upper quartile: Q3 = 50
Maximum: 56
Interquartile range: IQR = 21
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Joseph paid $105,000 for his home twenty years ago. since then, his house has increased its property value by 2.0% every year. in addition, joseph has made renovations and improvements to the house which will increase its sale value by $28,700. if joseph sells his home, how much profit will he make, to the nearest hundred dollars? a. $79,700 b. $84,700 c. $93,700 d. $122,400
If joseph sells his home, profit will he make = (a) $79720
What is profit and its example?
Profit is a term that often describes the financial gain a business receives when revenue surpasses costs and expenses. For example, a child at a lemonade stand spends one quarter to create one cup of lemonade. She then sells the drink for $2. Her profit on the cup of lemonade amounts to $1.75.Profit earned on the house if he sells = $79720
Amount paid by Joseph for his home = $105000
His house has increased its property value by 2.0% every year
So, Property cost after 20 years :
[tex]Cost = 105000 * (1.02)^{20}[/tex]
Cost = 105000 × 1.4859
Cost = 156020
Also, Due to renovations and improvements to the house, the property cost is increased by $28700
So, Total cost of the house after 20 years = 156020 + 28700
= $184720
So, Profit earned on the house if he sells = 184720 - 105000
= $79720
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Answer:a) $79720
Step-by-step explanation:edge
If the area of the given trapezoid is 1,188m^2, find the value of x.
Answer: x = 52 m
Step-by-step explanation:
The formula we will use, which solves for the area of a trapezoid, is [tex]a=\frac{a+b}{2} h[/tex] where a is a base, b is a base, and h is the height.
We will input the known values, then solve for x.
[tex]\displaystyle a=\frac{a+b}{2} h[/tex]
[tex]\displaystyle 1,188\;\text{m}^2=\frac{36\;\text{m}\;+x}{2}*27\;\text{m}[/tex]
[tex]\displaystyle 1,188\;\text{m}^2=(18\;\text{m}+\frac{x}{2})*27\;\text{m}[/tex]
[tex]\displaystyle 1,188\;\text{m}^2=486\;\text{m}+\frac{27x}{2}[/tex]
[tex]\displaystyle 702\;\text{m}^2=\frac{27x}{2}[/tex]
[tex]\displaystyle 1,404\;\text{m}^2=27x[/tex]
[tex]\displaystyle x=52\;\text{m}[/tex]
18 PTS!! What is the 400th term of the sequence below when using the explicit formula
aₙ=a₁+(n-1)•d?
79, 82, 85, 88,...
A. 1273
B. 1279
C. 1276
D. 1282
Answer:
[tex]\huge\boxed{\sf a_{400}=1276}[/tex]
Step-by-step explanation:
Sequence:79, 82, 85, 88, ....
Explicit formula:[tex]a_n=a_1+(n-1)d[/tex]
Where,
[tex]a_n = a_{400[/tex] (the term that is to be found)[tex]a_1 = 79[/tex] (the first term)[tex]n = 400[/tex][tex]d = 3[/tex] (difference between first and second term)Put the givens in the above formula
[tex]a_{400}=79 +(400-1)3\\\\a_{400}=79 + 399\times3\\\\a_{400}=79+ 1197\\\\a_{400}=1276\\\\\rule[225]{225}{2}[/tex]
What is the domain of the function graphed below? x<7 x less than or equal to 7 -2 less than or equal to x less than or equal to 3 all real numbers
The domain of the function shown in the graph is x < 7
How to determine the domain of the function shown in the graph?The complete question is added as an attachment
The domain is the set of x values the function can take
From the graph, we have the following x values
Minimum x value = Negative infinityMaximum x value = 7 (with an open circle)This means that the domain of x in the graph is x < 7
Hence, the domain of the function shown in the graph is x < 7
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In how many ways can you select five people from a group of ten if the order of selection is important?
There are 30240 ways in which we can select five people from a group of ten if the order of selection is important.
Given that there are 10 total number of people.
We are required to find the number of ways in which 5 people can be selected from 10 people.
Permutations is basically the number of arrangements that can be possible with the number of things , people, etc. It is denoted as [tex]nP_{r}[/tex]=n!/(n-r)!.
We have 10 people and we have to select 5 people from them then the number of ways can be find out by [tex]10P_{5}[/tex].
We have to expand this permutation.
=10!/(10-5)!
=10!/5!
=10*9*8*7*6*5!/5!
=10*9*8*7*6
=30240 ways
Hence there are 30240 ways in which we can select five people from a group of ten if the order of selection is important.
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need some help with this
Answer:XDDXXDXD
DXDXXDXDDXDXDXDXXDDXDEDXXDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
Step-by-XDDXDXDXstep explanation:
seria tu cacá XDXDXDXDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
5. Mr. Griffin asked the students to form the number eighty-four million, five
hundred thirty thousand, two hundred sixteen. Ben still had the number card
with the 3. In what place was Ben standing? Show your work in the space
below. Remember to check your solution
Answer:
Ben was standing in the
Step-by-step explanation:
84 000 000
+ 00 530 000
+ 00 000 216
_________________
84 530 216
From this u can see that the 3 is in the ten thousands place
Hello its urgent i really need help its a math question! i will give brainlist thank you
Find the area of a triangle with base of 10 inches and altitude to the base of 16 inches. 135 in 2 160 in 2 80 in 2
For a triangle with base of 10 inches and altitude of 16 inches, its area will be 80 in² (third option).
Calculation For the Area of the Triangle:
It is given that,
The base of the triangle, B = 10 inches
The altitude from the base of the triangle or height, H = 16 inches
Now, the formula used for finding the area of a triangle is given as follows,
Area, A = (1/2) × B × H
Substituting the given values of B and H in the above formula for area, we get,
A = (1/2) × (10 inches) × (16 inches)
A = 5 × 16 inches²
A = 80 inches²
Thus, the area of the triangle have base 10 inches and altitude 16 inches is 80 square inches.
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On a coordinate plane, parallelogram A B C D has points (3, 6), (6, 5), (5, 1), and (2, 2).
What is the area of parallelogram ABCD?
13 square units
14 square units
15 square units
Answer:
13 square units
Step-by-step explanation:
We need to find the distance between the points using the
Pythagorean Theorem, a² + b² = c²
(3,6) and (6,5); 6 - 3 = 3 and 6 - 5 = 1
3² + 1² = c²
9 + 1 = c²
c = √10
(6,5) and (5,1); 6 - 5 = 1 and 5 - 1 = 4
1² + 4² = c²
1 + 16 = c²
c = √17
√17 + √10 = 13.038
Please helppp!! i will hand brainlist and 19 points aswell thank you :)
I am not sure but when I calculated answer is √52...........
Answer:
[tex]\sqrt{52}[/tex]
Step-by-step explanation:
You would use the Pythagorean Thereon to find the answer.
[tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex] You can You can draw a right triangle from the two points. One way you could do this is draw a line straight down from N and stop when you get down to N'. Then continue drawing the line horizontally until it gets to N'. Connect N' to N and will have drawn a right triangle.
The vertical height of your triangle would be the difference of you y values and the horizontal distance would be the difference between your x values.
9 - 3 = 6
6 - 2 = 4
[tex]6^{2}[/tex] + [tex]4^{2}[/tex] =[tex]c^{2}[/tex]
36 + 16 = [tex]c^{2}[/tex]
52 = [tex]c^{2}[/tex]
[tex]\sqrt{52}[/tex] = c
Which expression represents the following statement? Multiply 6 by 3, and then subtract the quotient of 18 and 9.
A) 6 x (3 ÷ 18) + 9
B) 6 + 3 + (18 ÷ 9)
C) (6 x 3) − 18 ÷ 9
D) 6 x (3 + 18) + 9
Answer:
C) (6 x 3) - 18 ÷ 9
Step-by-step explanation:
I'll solve this question by writing down every step and doing it one by one.
1. Multiply 6 by 3
--> This will be represented as 6 x 3.
2. Subtract the quotient of 18 and 9
--> The quotient is a number resulting in a division of another number. In this case, 18 is bigger than 9, so the quotient will be: 18 ÷ 9.
--> Since we are subtracting this from the equation we got earlier, it can be expressed as (6 x 3) - (18 ÷ 9)
This equation is the same as C) (6 x 3) - 18 ÷ 9, so C is our final answer.
what is the value of x= and y=
Answer:
x = 18
y = 18
Step-by-step explanation:
We are given a right triangle (notice the 90 degree angle), with one of the angles being 45°.
We also know that the hypotenuse (the side opposite of the 90° angle) is 18√2.
We want to find the value of x and y.
First, let's figure out what the value of the other angle is.
The angles in a triangle all add up to 180 degrees.
Let's call the value of the angle we don't know s.
s + 45 + 90 = 180
Add 45 and 90 together.
s + 135 = 180
Subtract 135 from both sides.
s = 45
So the value of the other angle is 45 degrees.
When a right triangle is a 45-45-90 triangle (the numbers referring to the measures of the degrees), there is actually something special about it; if the legs (the sides that make up the 90 degree angle) have the value a (they would both be congruent because a 45-45-90 triangle is isosceles), then the hypotenuse will be a√2.
Keeping this in mind, let's solve for a.
Set 18√2 equal to a√2.
18√2=a√2
Divide both sides by √2.
18 = a
The value of a, aka the legs is a.
Both x and y would be equal to a in this case.
Therefore, x = 18, and y = 18.
Evaluate the following integral or state that it does not exist (Calculus 2) Please show step by step explanation!
Answer:
[tex]\displaystyle \int^1_0 \dfrac{e^{\sqrt{x}}}{\sqrt{x}}\:\:\text{d}x=2(e-1)[/tex]
Step-by-step explanation:
Given definite integral:
[tex]\displaystyle \int^1_0 \dfrac{e^{\sqrt{x}}}{\sqrt{x}}\:\:\text{d}x[/tex]
Integration by substitution:
Substitute u for one of the functions to give a function that's easier to integrate.
[tex]\textsf{Let }u=\sqrt{x}[/tex]
Find the derivative of u and rewrite it so that dx is on its own:
[tex]\implies \dfrac{\text{d}u}{\text{d}x}=\dfrac{1}{2\sqrt{x}}[/tex]
[tex]\implies \text{d}x=2 \sqrt{x}\:\text{d}u}[/tex]
[tex]\implies \text{d}x=2u\:\text{d}u[/tex]
Use the substitution to change the limits of the definite integral from x-values to u-values:
[tex]\textsf{When }x=0 \implies u=\sqrt{0}=0[/tex]
[tex]\textsf{When }x=1 \implies u=\sqrt{1}=1[/tex]
Therefore, the limits are unchanged.
[tex]\boxed{\begin{minipage}{3 cm}\underline{Integrating $e^x$}\\\\$\displaystyle \int e^x\:\text{d}x=e^x+\text{C}$\end{minipage}}[/tex]
Substitute everything into the original integral and solve:
[tex]\begin{aligned}\implies \displaystyle \int^1_0 \dfrac{e^{\sqrt{x}}}{\sqrt{x}}\:\:\text{d}x & =\displaystyle \int^1_0 \dfrac{e^u}{u} \cdot 2u\:\:\text{d}u \\\\ & = \displaystyle \int^1_0 2e^u\:\:\text{d}u \\\\ & = 2 \int^1_0 e^u\:\:\text{d}u\\\\ & = 2 \left[ e^u\right]^1_0\\\\ & = 2 \left[e^1-e^0\right]\\\\& = 2(e-1)\end{aligned}[/tex]
Learn more about definite integrals here:
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