Answer: The antiderivative of [tex]e^{x^{2}}[/tex] is [tex]\sqrt{\pi} \sqrt{e^{x^2}-1} +C[/tex].
Explanation:
[tex]I=\int{e^{x^2}dx}\\I^2=\int{e^{x^2}dx}\int{e^{x^2}dx}[/tex]
Using dawson integral.
[tex]I^2=\int{e^{x^2}dx}\int{e^{y^2}dy}[/tex]
[tex]I^2=\int{\int{e^{x^2+y^2}dxdy}}[/tex]
Put [tex]x^2+y^2=r^2[/tex]
[tex]I^2=\int{\int{e^{r^2}rdrd\theta}}[/tex]
[tex]\int_{0}^{2\pi}{d\theta}\int_{0}^{x}{re^{r^2}dr}[/tex]
Use substitution method and sunbstitute [tex]r^2=t[/tex].
[tex]\int_{0}^{2\pi}{d\theta}\int_{0}^{x^2}{\frac{1}{2}e^tdt}[/tex]
[tex]I^2=\frac{1}{2}|\theta|_{0}^{2\pi}|e^t|_{0}^{x^2}\\I^2=\frac{1}{2}(2\pi)(e^{x^2}-1)\\I=\sqrt{\pi}\sqrt{e^{x^2}-1}[/tex].
Therefore, the antiderivative of [tex]e^{x^{2}}[/tex] is [tex]\sqrt{\pi} \sqrt{e^{x^2}-1} +C[/tex].
Two sides of an obtuse triangle measure 10 inches and 15 inches. The length of longest side is unknown. What is the smallest possible whole number length of the unknown side?
Answer:
24 inches.
Step-by-step explanation:
Two sides of an obtuse triangle measure 10 inches and 15 inches.
The length of the longest side is unknown.
We have to find the smallest possible whole number for length of the unknown side.
Since a triangle is possible only when sum of two sides of the triangle is greater than the third side.
Therefore, length of third side < sum of other two sides
length of third side < 10 + 15
length of third side < 25
Therefore smallest whole number length should be 24 inches.
Which of the following fractions is written in its simplest form
13/25,
any time you see a prime number in the numerator, its in its simplest form.
For this case we have the following fractions:
22/33 42/81 13/25 18/27
Fraction 1: You can divide the numerator and denominator by 11
[tex]\frac{22}{33}= \frac{2}{3}\\[/tex]
Fraction 2: The numerator and denominator can be divided by 3
[tex]\frac{42}{81}= \frac{14}{27}\\[/tex]
Fraction 3: The numerator and denominator can not be divided by the same number.
[tex]\frac{13}{25}\\[/tex]
Fraction 4: The numerator and denominator can be divided by 9
[tex]\frac{18}{27}= \frac{2}{3}\\[/tex]
Answer:
A fraction that is written in its simplest form is
Fraction 3: [tex]\frac{13}{25}\\[/tex]
A rectangle is placed inside a circle whose diameter is 13 with all of its corners on the circle. The area of the rectangle is 60. What are the dimensions of the rectangle? Hint: Draw a picture and use the Pythagorean Theorem.
Given
A rectangle is placed inside a circle whose diameter is 13unit all corners of it on the circle.
area of the rectangle = 60 square unit
Find the dimensions of the rectangle.
To proof
As given in the question
rectangle is placed inside a circle whose diameter is 13unit .
now by using the diagram given below
the diagonal of the rectangle is equal to the diameter of the circle.
Thus DB = 13 unit
FORMULA
Area of the rectangle = Length × Breadth
area of the rectangle = 60 square unit
let the value of the length be ( Say AB ) = x
put the value in the above equation
we get
60 = x × Breadth
[tex]Breadth( say AD)= \frac{60}{x}[/tex]
now in ABD
Using the pythagorus theorem
we get
BD²= AB² + AD²
[tex]13^{2} = x^{2}+ (\frac{60}{x}) ^{2}[/tex]
solving the above equation
we get
[tex]169x^{2}= x^{4} + 3600[/tex]
[tex]x^{4} - 169x^{2}+3600 =0[/tex]
[tex]x^{4} - 144x^{2} -25x^{2} + 3600 =0[/tex]
[tex]x^{2} (x^{2} - 144) - 25 (x^{2} - 144) =0[/tex]
(x² - 25) (x² -144) =0
The roots are
x = -12 ,-5, 5, 12
-12 and -5 are neglected because of negative terms .
( Length and breadth of recangle cannot be negative )
thus x = 12 and 5
Two case arise
First case
when
Length of rectangle= 12unit
[tex]Breadth = \frac{60}{12}[/tex]
Breadth of rectangle = 5 unit
Second case
Length of rectangle= 5unit
[tex]Breadth = \frac{60}{5}[/tex]
Breadth of rectangle = 12 unit
Hence proved
Final answer:
To find the dimensions of a rectangle inside a circle with a diameter of 13, we use the Pythagorean Theorem and the area of the rectangle. After substituting values and solving, we find the dimensions are 5 units by 12 units.
Explanation:
To find the dimensions of the rectangle that fits inside a circle with a diameter of 13, one must first recognize that the rectangle forms the sides of a right-angled triangle where the diagonal of the rectangle serves as the hypotenuse of the triangle. This hypotenuse, thus, equals the diameter of the circle, which is 13 units.
Since the area of the rectangle is 60 square units, we let the sides of the rectangle be denoted as length (l) and width (w), with the area equation being l×w = 60.
Using the Pythagorean Theorem, we have: l2 + w2 = 132. We also know that lw = 60, which allows us to solve for one variable in terms of the other. Let's solve for w: w = 60/l. Substituting this into the Pythagorean equation gives l2 + (60/l)2 = 169. Solving this quadratic equation will give us the dimensions of the rectangle that can fit within the circle.
After solving, one finds that the dimensions that satisfy both the area and the Pythagorean Theorem are 5 and 12.
Therefore, the dimensions of the rectangle are 5 units by 12 units.
Squaring a number and subtracting 6 gives the same result as doubling the number and adding 9.
The answer is 5.
5^2 = 25 25 - 6 = 19
5 x 2 = 10 10 + 9 = 19
A map of a town has no scale. You know that the distance between the bus stop and the park is 6 miles. On the map it is 3 inches. Also, on the map you know the distance from the park to the house is 8 inches. What is the actual distance in miles between the park and the house?
Answer:
The answer is 16
Resolving inequalities 6/5x-12<2-3/2x
Answer: x < [tex]\frac{140}{27}[/tex]
Step-by-step explanation:
[tex]\frac{6}{5}x - 12 < 2 - \frac{3}{2}x[/tex]
[tex]10(\frac{6}{5}x - 12) < 10(2 - \frac{3}{2}x)[/tex]
2(6)x - 10(12) < 10(2) - 5(3)x
12x - 120 < 20 - 15x
+15x +15x
27x - 120 < 20
+120 +120
27x < 140
[tex]\frac{27x}{27} < \frac{140}{27}[/tex]
x < [tex]\frac{140}{27}[/tex]
23rd term: -4,-7,-10,-13
Determine how much the bill will be if there are 50 text messages
Answer:
about 105
Step-by-step explanation:
Violin strings are parallel. Viewed from above, a violin bow in two different positions forms two transversals to the violin strings. Find x and y in the diagram.
ANSWER
[tex]x=10[/tex]
and
[tex]y=20[/tex]
EXPLANATION
Since corresponding angles are equal,
[tex]4x+y=60---(1)[/tex]
Alternate angles are also equal, therefore,
[tex]8x+y=100---(2)[/tex]
Equation (2) minus equation (1) gives,
[tex]\Rightarrow 4x=40[/tex]
[tex]\Rightarrow x=10[/tex]
We put [tex]x=10[/tex] in to equation (1)
[tex]4(10)+y=60[/tex]
[tex]y=60-40[/tex]
[tex]y=20[/tex]
To find x and y in the violin string diagram, use the properties of angles formed by transversals and parallel lines. Corresponding angles and interior angles are key relationships. In the example, x = 15° and y = 90°.
The problem involves understanding the relationships between angles formed by transversals and parallel lines. When a violin bow is placed across parallel violin strings, it forms transversals that create various angles.
Given that the violin strings are parallel, the interior angles formed on the same side of the transversal are supplementary. Let's consider the setup of the diagram:
Identify corresponding angles, alternate interior angles, and alternate exterior angles.Use the properties of parallel lines and transversals to set up equations.Solve for the unknowns (x and y).Example: Suppose the violin strings and the bow are positioned in such a way that we can see the angles formed. Assume the given angles are as follows:
Angle 1 = 30° (corresponding angle)Angle 2 = (2x)° (alternate interior angle)Angle 3 = y°Step 1: Since Angle 1 and Angle 2 are corresponding angles and lie between parallel lines:
30° = 2x → 2x = 30° → x = 15°
Step 2: To find y, use the fact that the sum of angles around a point is 360°. Suppose y is opposite to the sum of Angle 1 and another given angle on the same side:
If the other angle on the same side is 60°:
y = 180° - (Angle 1 + 60°) → y = 180° - (30° + 60°) → y = 90°
Thus, the values of x and y are determined to be 15° and 90°, respectively.
I will love you forever if someone can help me with this
Geometry PLEASE HELP ASAP! Randy wants to attach a 14 foot string of lights to the top of the 12 foot mast of his sailboat. How far from the base of the mast should he attach the end of the light string? Round your answer two decimal places.
The answer is 7.21.
Randy should attach the end of the light string approximately 7.21 feet from the base of the mast using the Pythagorean theorem, as the distance from the base to the end of the string forms the leg of a right-angled triangle with the mast and the string itself.
Explanation:To solve Randy's problem, we need to use the Pythagorean theorem, which is applicable because we are dealing with a right triangle formed by the mast, the ground, and the string of lights. The mast is one leg of the triangle, and the string of lights is the hypotenuse. We are trying to find the length of the other leg, which is the distance from the base of the mast to the point where the end of the light string will be attached.
The formula for the Pythagorean theorem is:
a² + b² = c²
Where:
a is one leg of the triangleb is the other leg of the trianglec is the hypotenuse (the side opposite the right angle)In this scenario, a is the mast with a known length of 12 feet, c is the string of lights with a known length of 14 feet, and b is the distance from the base of the mast we need to calculate.
By substituting the known values into the theorem, we have:
12² + b² = 14²
Which simplifies to:
144 + b² = 196
Subtracting 144 from both sides:
b² = 196 - 144
b² = 52
Then, take the square root of both sides:
b = √52
After calculating, we find that:
b ≈ 7.21 feet
So Randy should attach the end of the light string approximately 7.21 feet from the base of the mast to reach the top.
math help please help... ive been stuck on this, thank you so much !!!!!
Answer: (-3,4) and (9,20)
I think these two points will make the answer you have been looking for.
WITH ANSWER
What is the exact value of x in the exponential equation 2(10)^3x=26 in log
*****x=log13/3
[tex]2\cdot 10^{3x}=26\\10^{3x}=13\\\log 10^{3x}= \log 13\\3x\log 10= \log 13\\3x= \log 13\\\\[/tex]
[tex]x=\frac{\log 13}{3}[/tex]
lmk if you have questions
To find the exact value of x in the exponential equation 2(10)^3x=26, divide both sides by 2, apply the common logarithm, use the power rule of logarithms, and simplify to find that x equals log(13)/3.
To solve the exponential equation 2(10)³ˣ=26 for the exact value of x using logarithms, follow these steps:
Divide both sides of the equation by 2 to isolate the exponential term:Thus, the exact value of x is log(13)/3.
simply each expression, use the distributive property to clear parentheses then combine like terms
6(x-2)+5x
Answer:
11x - 12
Step-by-step explanation:
Distributive property, also called the distributive law of multiplication and division, lets us distribute a number by multiplying it to each of the numbers in a term separately.
We can expand this expression by clearing the parentheses and combining the like terms here by applying the distributive property:
6 (x-2) + 5x
6x - 12 + 5x ---> (6 is multiplied to both the terms, x and -2, separately)
Now we will combine like terms to get:
6x + 5x - 12
11x - 12
What is the product of 3a + 5 and 2a +4a - 2
The product of the expressions 3a + 5 and 6a - 2 is 18a² + 24a - 10. The distributive property is used to multiply each term in the first expression by each term in the second.
The product of the two expressions 3a + 5 and 2a + 4a - 2 can be found using the distributive property. First, combine like terms in the second expression to simplify it to 6a - 2. Now, to find the product, multiply each term in the first expression by each term in the simplified second expression:
Multiply 3a by 6a to get 18a².
Multiply 3a by -2 to get -6a.
Multiply 5 by 6a to get 30a.
Multiply 5 by -2 to get -10.
Combining these products gives you the final answer: 18a² + 24a - 10.
An edge on a cube is the line segment joining two vertices. What is the sum of the lengths of the edges if each edge measures 12 + 10V6 inches? Write your answer as a radical expression.
which of the following doesNOT represent a rational number
HURRY PLEASE
A) 0
B)2 1/2
C)square root of 3
D) -1/10
C. The square root of 3 is an irrational number. I hope that this helps!
Are angles 1 and 4 linear pair, vertical angles or neither? Thank you! :)
Answer:
it is a linear one i think
The classification of angles 1 and 4 as either linear pairs, vertical angles or neither is impossible without more context or a diagram. Linear pairs are adjacent angles that form a straight line and total 180 degrees, while vertical angles are directly opposite each other from an intersection of two lines and are equal.
Explanation:Without a proper diagram or more context, it's challenging to determine whether angles 1 and 4 are linear pairs, vertical angles, or neither. However, I can explain what these terms mean. Linear pairs are adjacent angles whose non-common sides form a straight line, adding up to 180 degrees. Vertical angles are angles opposite each other when two lines intersect. They are equal in measure. Unfortunately, without a specific diagram indicating the positions of angles 1 and 4, I can't definitively answer your question.
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What’s is your average speed in miles per hour and in feet per second if you travel a mile in 26 minutes?
To calculate the average speed, divide the total distance travelled by the total time taken. The speed is approximately 2.31 mph and 3.385 fps.
Explanation:To calculate the average speed you divide the total distance travelled by the total time taken. In this case, the distance is one mile and the time is 26 minutes.
To get the speed in miles per hour (mph), first convert the time from minutes to hours by dividing the time in minutes by 60. So, 26 minutes is approximately 0.43 hours. Then, divide the distance by the time to get the speed. Therefore, your speed is roughly 2.31 mph.To convert this to feet per second (fps), first convert 1 mile, which is 5280 feet. Then convert the time from 26 minutes to seconds, which is 1560 seconds.So, when you divide distance by time, you get approximately 3.385 fps.
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What is 7.2 times 10 to the 7th power times 14.6 x 10 to the 5th power?
7 x [tex]10^{7}[/tex] times 14.6 x[tex]10^{5}[/tex]
Multiply 7 and 14.6 and add exponents
Final Answer 102.2 and [tex]10^{5}[/tex]
hope that helps :)
7.2 x 10 x 5 x 14.6 x 10 x 5
7.2 x 100000 x 14.6 x 100000
720000 x 1460000 = 1.0512 x 10 x 12
GameStop has the Xbox one advertised for 15% off the original price of $350 plus 6% sale tax online Xbox one is advertised at Best Buy for $310.00 plus 9% sales tax which is the better buy
GameStop's total will be 371.00
Best Buy's total will be 337.90
Best Buy will be the better buy
Your answer is Best Buy
Answer:
The best option for buying the Xbox One is GameStop, as the final price would be of $315.35.
Step-by-step explanation:
GameStop: To obtain the price in GameStop, we must first obtain the 15% discount on the price of 350, and subtract it from the initial price. Then, we must calculate the amount of the tax and add it to the price to obtain the final price.
(350 - (350 x 0.15)) x 1.06 = $315.35
Best Buy: Instead, at Best Buy we simply add the sales price tax.
310 x 1.09 = $337.90
A building in a downtown business area casts a shadow that measures 88 meters along the ground. The straight-line distance from the top of the building to the end of the shadow it creates is at a 32° angle with the ground. What is the approximate height of the building? Round your answer to the nearest meter.
Answer: 55 meters
Step-by-step explanation:
Let the height of the building be 'h'.
Since building is standing vertical to the ground making right angle.
Then , the triangle formed is a right triangle .
The given angle of elevation : [tex]32^{\circ}[/tex]
Using trigonometry , we have
[tex]\tan\theta=\dfrac{\text{Perpendicular}}{\text{Base}}\\\\\tan(32^{\circ})=\dfrac{h}{88}\\\\\Rightarrow\ 0.6248693519=\dfrac{h}{88}\\\\\Rightarrow\ h=0.6248693519\times88=54.988502967\approx55\text{ meters}[/tex]
Hence, the approximate height of the building = 55 meters
will give brainliest
Answer:
12
Step-by-step explanation:
2 times 2 is 4
4 divided by 2 is 2
2 times 2 is 4
2 times 6 is 12
12 divided by 2 is 6
4 plus 2 plus 6 is 12
which is the simplified form of (9c^-9)-3
3[3c^(-9)-1]
hope this helps
Answer:
b
Step-by-step explanation:
how do i prove that measurement of angle abc is equal to measurement of def
What is the simplified product of (2x)(x^2)+(2x)(x)+(2x)(-2)+(3)(x^2)+(3)(x)+(3)(-2)
(2x)(x^2)+(2x)(x)+(2x)(-2)+(3)(x^2)+(3)(x)+(3)(-2)
Simplifying
(2x)(x2) + (2x)(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)
Remove parenthesis around (2x)
2x(x2) + (2x)(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)
Multiply x * x2
2x3 + (2x)(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)
Remove parenthesis around (2x)
2x3 + 2x(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)
Multiply x * x
2x3 + 2x2 + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)
Remove parenthesis around (2x)
2x3 + 2x2 + 2x(-2) + (3)(x2) + (3)(x) + (3)(-2)
Reorder the terms for easier multiplication:
2x3 + 2x2 + 2 * -2x + (3)(x2) + (3)(x) + (3)(-2)
Multiply 2 * -2
2x3 + 2x2 + -4x + (3)(x2) + (3)(x) + (3)(-2)
Multiply 3 * -2
2x3 + 2x2 + -4x + 3x2 + 3x + -6
Reorder the terms:
-6 + -4x + 3x + 2x2 + 3x2 + 2x3
Combine like terms: -4x + 3x = -1x
-6 + -1x + 2x2 + 3x2 + 2x3
Combine like terms: 2x2 + 3x2 = 5x2
-6 + -1x + 5x2 + 2x3
Answer:
The product is,[tex]-6 + -1x + 5x2 + 2x3[/tex]
explanation:
Simplifying
[tex](2x)(x2) + (2x)(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Remove the parentheses (2x)
[tex]2x(x2) + (2x)(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Multiply [tex]x * x2[/tex]
[tex]2x3 + (2x)(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Remove the closing parentheses (2x)
[tex]2x3 + 2x(x) + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Multiply [tex]x * x[/tex]
[tex]2x3 + 2x2 + (2x)(-2) + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Remove all parentheses (2x)
[tex]2x3 + 2x2 + 2x(-2) + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Rearrange the terms to make multiplication easier:
[tex]2x3 + 2x2 + 2 * -2x + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Multiply [tex]2 * -2[/tex]
[tex]2x3 + 2x2 + -4x + (3)(x2) + (3)(x) + (3)(-2)[/tex]
Multiply [tex]3 * -2[/tex]
[tex]2x3 + 2x2 + -4x + 3x2 + 3x + -6[/tex]
Reorder the terms:
[tex]-6 + -4x + 3x + 2x2 + 3x2 + 2x3[/tex]
Combine like terms: -4x + 3x = -1x
[tex]-6 + -1x + 2x2 + 3x2 + 2x3[/tex]
Combine like terms: 2x2 + 3x2 = 5x2
[tex]-6 + -1x + 5x2 + 2x3[/tex]
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william walked down a mountain for a total of -200 feet. He traveled this distance for 20 minutes. How far had william traveled in 1 minute?
A. 400 feet
B. 10 feet
C.-10 feet
D.-400 feet
Answer:
-200 feet/20 minutes = -10 feet/minute
The correct answer is C.
Help find the measure of b
b= 81 degrees
Explanation:
A straight line is 180 degrees.
180 - 99 = 81
All the angles combined should be equal to 360, forming a circle.
In order to find angle b, we have to know what supplementary angles.
Supplementary angles are two angles that add up to 180°, or in other words, a straight line.
Angle b and the 99 degree angle are supplementary due to them creating a straight line when combined, so they add up to 180°.
To find angle b, subtract 99 degrees from 180.
180 - 99 = 81
Angle b is equal to 81 degrees.
How does 15.050 compare to 15.500
15.050 < 15.500
notice that 500 is more than 50... so hope this helps a little bit !!!
:)
Answer:
15.050 < 15.500
Step-by-step explanation:
15.050 compare to 15.500
WE need to compare both the decimal numbers
To compare the numbers we compare each digit from left to right
1 is equal to 1
5 is equal to 5
now we compare the number after the decimal point
0 is less than 5 (the numbers in tenths place)
15.050 is less than 15.500
15.050 < 15.500
Antonio graphs these equations and finds that the lines intersect at a single point, (–5, 0.25). Equation A: Equation B: 4y – 3x = 16 –x – 8y = 3 Which statement is true about the values x = –5 and y = 0.25?
Given equations : Equation A: 4y -3x =16 and
Equation B: -x-8y =3.
We are given a point (-5,0.25) plugging it in first equation
4(0.25) -3(-5) =16
1+15 =16.
16 =16 : Satisfies first equation.
Plugging (-5,0.25) it in second equation.
-(-5)-8(0.25) =3.
5 - 2 = 3.
3=3 : Satisfies first equation.
Therefore, they are the only values that make both equations true.