Consider the prime factorization of 20!.
[tex]20! = 20 \times 19 \times 18 \times \cdots \times 3 \times 2 \times 1[/tex]
The LCM of 1, 2, ..., 20 must contain all the primes less than 20 in its factorization, so
[tex]M = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times m[/tex]
where [tex]m[/tex] is some integer not divisible by any of these primes.
Compare the factorizations of the remaining divisors of 20!, and check off any whose factorizations are already contained in the product of primes above.
[tex]4 = 2^2[/tex] - missing a factor of 2
[tex]6 = 2\times3[/tex] - ✓
[tex]8 = 2^3[/tex] - missing a factor of 2²
[tex]9 = 3^2[/tex] - missing a factor of 3
[tex]10 = 2\times5[/tex] - ✓
[tex]12 = 2^2\times3[/tex] - missing a factor of 2
[tex]14 = 2\times7[/tex] - ✓
[tex]15 = 3\times5[/tex] - ✓
[tex]16 = 2^4[/tex] - missing a factor of 2³
[tex]18 = 2\times3^2[/tex] - missing a factor of 3
[tex]20 = 2^2\times5[/tex] - missing a factor of 2
From the divisors marked "missing", we add the necessary missing factors to the factorization of [tex]M[/tex], so that
[tex]M = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 2^3 \times 3[/tex]
Then the LCM of 1, 2, 3, …, 20 is
[tex]M = 2^4 \times 3^2 \times 5 \times7 \times 11 \times 13 \times17 \times 19[/tex]
[tex]\implies \boxed{M = 232,792,560}[/tex]
What line is parallel to the equation y=5x+1
A rocket is launched at the top of a building. The height of the rocket (in meters) written
in terms of time (seconds) can be modeled by h (t) = -4.9t2 +9.8t + 73.5. What is
the maximum height?
The maximum height of the rocket is 78.4m
Maximum height of a functionGiven the function that represents the height of a function expressed as:
h(t) = -4.9t² +9.8t + 73.5.
The rocket will reach its maximum height at the point where h'(t) = 0
h'(t) = -9.8t +9.8
Equate to zero
0 = -9.8t + 9.8
9.8t = 9.8
t = 1
Substitute
h(1) = -4.9(1)²+9.8(1) + 73.5
h(1) = 4.9 + 73.5
h(1) = 78.4m
Hence the maximum height of the rocket is 78.4m
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Fatima evaluated the expression StartFraction 4 m Superscript negative 3 Baseline n Superscript negative 2 Baseline Over m Superscript negative 1 Baseline n EndFraction, when m = negative 2 and n = 4. Her work is shown below.
StartFraction 4 m Superscript negative 3 Baseline n Superscript negative 2 Baseline Over m Superscript negative 1 Baseline n EndFraction = 4 m Superscript negative 2 Baseline n Superscript negative 3 Baseline = 4 (negative 2) Superscript negative 2 Baseline (4) Superscript negative 3 Baseline = StartFraction 1 Over 64 EndFraction times StartFraction 1 Over 64 EndFraction = StartFraction 1 Over 4,096 EndFraction
What was Fatima’s error?
She subtracted the exponents incorrectly when simplifying the expression.
She substituted the wrong values for the variables.
She applied the exponent Negative 2 to 4 (negative 2) instead of applying the exponent to just Negative 2.
She found an incorrect value for (Negative 8) Superscript negative 2 since the value should be negative.
The error of Fatima is: C. She applied the exponent -2 to "4(-2) instead of only to "(-2).
What is an Exponent?An exponent in any expression is the letter or number on the right of a base in an expression.
In the evaluated expression shown, the base that the exponent "-2" applies to is "m", and does not include 4. If we substitute the value of m = -2, into the evaluated expression, the base that the exponent "-2" applies to is "-2" and does not include "4".
Therefore, Fatima's mistake is: C. She applied the exponent -2 to "4(-2) instead of only to "(-2).
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A rectangle has a height of 7x^27x 2 7, x, squared and a width of 3x^2+8x+73x 2 +8x+73, x, squared, plus, 8, x, plus, 7. Express the area of the entire rectangle.
The expression 21x⁴ + 56x³ + 49x² represents the area of the entire rectangle.
What is the area of the rectangle?It is defined as the area occupied by the rectangle in two-dimensional planner geometry.
The area of a rectangle can be calculated using the following formula:
Rectangle area = length x width
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
The area of the rectangle = length x width
= (3x² + 8x + 7)(7x²)
[tex]\rm =7x^2\cdot \:3x^2+7x^2\cdot \:8x+7x^2\cdot \:7[/tex]
= 21x⁴ + 56x³ + 49x²
Thus, the expression 21x⁴ + 56x³ + 49x² represents the area of the entire rectangle.
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Which triangles illustrate the Side Angle Side Similarity theorem (abbreviated S A S).
Answer:
Triangles a, b, and d
Step-by-step explanation:
The three triangles have a side length measurement, an angle measure, then another side length measurement
in that order (the order matters)
Triangle a,b and d are the answers.
How to find the side angle side (SAS)?"SAS" is when you know the two sides and the angle between them. Use the cosine theorem to calculate the unknown edge, then use the sine and cosine theorem to find the smaller of the other two angles, and then use the three angles that are 180 ° to find the last angle. .. The side of the
angle refers to the two rays or line segments that form the angle. In the figure below, the rays BA and BC are the sides of the angle ABC. An angle is formed by rotating a ray around its endpoint.
Edge-side assumptions-> If the two sides and edges of a triangle are congruent with the two sides and the edges of another triangle, the two triangles are congruent by the side-angle assumptions.
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find the value of x
Answer:
70
Step-by-step explanation:
What are the legnths of SV and QT
An equation is formed of two equal expressions. The length of the sides QT and SV are 21 units and 41 units respectively.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
Given the length of TR is 17 and it is equal to side VR, therefore,
VR = VT
3x + 2 = 17
x = 5
Since the given figure is a kite, therefore,
QT= QV = 4x + 1
QT = 4(5) + 1
QT = 21 units
Similarly, the sides SV will be equal to ST,
SV =ST = 9x - 4
SV = 9(5) - 4
SV = 41 units
Hence, the length of the sides QT and SV is 21 units and 41 units respectively.
The complete question is:
The length of TR is 17 units. What are the lengths of SV and QT?
SV=___ units
QT=___ units
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Using index notation , express the following number as products of their prime factors . (1) 45 (2) 80 (3) 420 (4) 625 (5) 81
3²×5, 2⁴×5, 2²×3×5×7, 5⁴ and 3⁴ are the index notations of the numbers 45, 80, 420, 625 and 81 respectively. This can be obtained by taking the factors of these numbers and wring as the powers as shown below.
What is index notation?
It is a method of representing numbers and variables that are multiplied by themselves a number of times.
For example, 7³, a²
What are the index notations of the given numbers?
The given numbers can be represented as the following:
(1) 45 =3×3×5 = 3²×5
(2) 80 =2×2×2×2×5 = 2⁴×5
(3) 420 =2×2×3×5×7 = 2²×3×5×7
(4) 625 =5×5×5×5 = 5⁴
(5) 81 =3×3×3×3 = 3⁴
Hence 3²×5, 2⁴×5, 2²×3×5×7, 5⁴ and 3⁴ are the index notations of the numbers 45, 80, 420, 625 and 81 respectively.
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I need to know the answer please help me
Directions: Indicate product of FOIL for the problem:
Answer:
[tex] {x}^{4} - {y}^{2} [/tex]
Step-by-step explanation:
This is a difference of squares, so its the square of the first term minus the square of the second term.
helpp quick pleaseee
Type the correct answer in the box. If necessary, use / for the fraction bar. A shaded triangular pyramid and an inverted shaded triangular pyramid has their bases on the bottom and the top sides of a rectangular prism. Both pyramids in the figure have the same base area as the prism. The ratio of the combined volume of the pyramids to the volume of the prism, expressed as a fraction in simplest form, is . Reset Next
The ratio of the combined volume of these pyramids to the volume of the prism as a fraction is equal to 1/3.
How to calculate the volume of a pyramid?Mathematically, the volume of a pyramid can be calculated by using this formula:
Volume = 1/3 × b × h
Where:
h is the height.b is the base area.Since both pyramids have the same base area as the prism, their combined volume is given by:
Combined volume = (1/3 × b × h) + (1/3 × b × h)
Combined volume = 2/3 × b × h
Also, the volume of this prism = 1/2 × b × h
Thus, the ratio is given by:
Ratio = (2/3 × b × h)/(1/2 × b × h)
Ratio = 2/3 × 1/2
Ratio = 1/3.
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Find the length of the short leg on the following right triangle.
60+90+ angle U = 180 {sum of angle of triangle}
or, 150+ U= 180
or, angle U = 30°
From here we got that ,
angle U is smallest.
We know that, side opposite to the shortest angle is shortest side. So , short leg is side PM
In right angled triangle PMU,
hypotenuse(h) = 10
angle theta = 60°
perpendicular (PM)= ?
NOW,
sin theta = Perpendicular/ hypotenuse
or, sin 60° = P/10
[tex] \\or \frac{ \sqrt{3} }{2} = p \div 10[/tex]
[tex] or\frac{10 \sqrt{3} }{2} = p[/tex]
[tex]or \: 5 \sqrt{3 } = p[/tex]
Therefore,
[tex]short \: leg = 5 \sqrt{3} [/tex]
4.
Please answer the question on exponential growth or decay. Round to
if
the nearest whole number or penny if the question is concerning money
necessary
Arubber bouncy ball is dropped from a
height of 192.00 cm onto a hard flat floor.
After each bounce, the ball returns to a
height that is 20% less than the previous
maximum height. What is the maximum
height reached after the 10th bounce?
Using a decaying exponential function, it is found that the maximum height reached after the 10th bounce is of 20.62 cm.
What is an exponential function?A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
A(0) is the initial value.r is the decay rate, as a decimal.In this problem, the parameters are:
A(0) = 192, r = 0.2.
Hence the maximum height after the nth bounce is:
[tex]A[n] = 192(0.8)^n[/tex]
After the 10th bounce, the maximum height in cm is of:
[tex]A[10] = 192(0.8)^10 = 20.62[/tex]
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What standard form polynomial expression represents the area of the triangle area = 1/2 bh
What is the probability of choosing a triangle?
Answer:
52.9%
Step-by-step explanation:
There are 17 shapes total. Nine of those shapes are triangles. Therefore the probability of choosing a triangle is 9/17.
To find that as a percentage type 9 divided by 17 into the calculator. The answer to that is 0.5294117647058824. To turn a decimal into a percentage move the decimal point two places over making the percentage: 52.94117647058824%.
Since the yellow text states to round the answer to the nearest hundredth of the percent the answer is 52.9%.
52.94%
Step-by-step explanation:
there are 17 objects and you divide the 9 triangles by 17 and get .529411765. Which is 52.94 percent.
A six-sided die is rolled. What are the odds that a number greater than 4 is rolled?
Answer: [tex]\frac{1}{3}[/tex] or about 33%
Step-by-step explanation:
We will divide the wanted outcomes by the possible outcomes.
[tex]\displaystyle \frac{\text{wanted outcomes}}{\text{possible outcomes}} =\frac{\text{Number of numbers greater than 4}}{\text{number of numbers 1 through 6}} =\frac{2}{6}=\frac{1}{3}\;\; \text{or about 33\%}[/tex]
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Answer: 1/3
Step-by-step explanation:
There are only 2 numbers greater than 4. 5 and 6.
2/6= 1/3 chance
Robyn, Rachel and Rocky bought a set of LEGO blocks, Robyn paid $45, Rachel paid $65 and Rocky paid $60. What
fraction of the total amount Rocky paid?
Answer:
6/17
Step-by-step explanation:
Robyn paid $45,
Rachel paid $65
Rocky paid $60.
Total amount paid = $45 + $65 + $60 = $170
Fraction paid by Rocky will be:
= 60/170
= 6/17
PLEASE LOOK AT THE PIC BELOW AND HELPPP
Answer: See attached.
Step-by-step explanation:
For [a], we graph the parent function (y = x²), but shift it downwards 2 units.
For [b], we graph the parent function (y = x²), but shift it downwards 6 units, shift it to the right 2 units, and apply a compress factor of 3.
I'm renting to own a house for $80,000. I paid down $6,000.00, leaving a balance of $74,000. For the past eight years I’ve paid 768.57 a month. How much have I paid from May 2014 to July 2022?
The total amount of money I have paid from May 2014 to July 2022 is $75,319.86.
How much I have paid?The first step is to determine the total number of months from May 2014 to July 2022.
12 months = 1 year
May 2022 - May 2014 = 8 years x 12 = 96 months
Number of months from may 2022 to july 2022 = 2 months
Total number of months = 96 + 2 = 98 months
Total amount paid = 98 x 768.57 = $75,319.86
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A linear function has a slope of Negative StartFraction 7 Over 9 EndFraction and a y-intercept of 3. How does this function compare to the linear function that is represented by the equation y + 11 = Negative StartFraction 7 Over 9 EndFraction (x minus 18)? a It has the same slope and the same y-intercept. b It has the same slope and a different y-intercept. c It has the same y-intercept and a different slope. d It has a different slope and a different y-intercept.
It has the same slope and the same y-intercept, that is, option a is the right choice.
A linear function in the slope-intercept is written as y = mx + b, where m represents the slope of the line and b represents the y-intercept.
We are given two linear functions and are asked to compare them.
First linear function:
Slope (m) = -7/9.
y-intercept (b) = 3.
Therefore, the linear function is: y = (-7/9)x + 3.
Second linear function:
Function given: y + 11 = (-7/9)(x - 18)
or, y + 11 = (-7/9)x + (-7/9)(-18)
or, y = (-7/9)x + 14 - 11 = (-7/9)x + 3.
Therefore, slope (m) = (-7/9)
y-intercept (b) = 3.
Therefore, we can say that it has the same slope and the same y-intercept, that is, option a is the right choice.
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Edward rewrote the product of two rational expressions to show the multiplication of the numerators and denominators. (5y/9)(x 3) (5y)(x 3)/(9)(4x)
If Edward rewrote the product of [tex](5y/9)(x//3)(5y)(x^{3}/9)(4x)[/tex] is [tex](100x^{5}y^{2} )/243[/tex].
Given Expressions [tex](5y/9)(x/3)(5y)(x^{3}/9)(4x)[/tex].
We have to solve this expression by finding product and product is the result of multiplication of two numbers.
Expression is a combination of numbers, symbols, variables, coefficients, fraction of numbers. It is not expressed in equal to form. It is generally evaluated to form full expression not to find the value of variables. Numerator is the number above division sign and number below the division sign is known as denominator.
(5y/9)(x/3)(5y)(x^{3}/9)(4x)=
We have to first add powers of same variables. and multiply the coefficients of all the variables whether they are in numerator and denominators.
=[tex](4x^{5} *25y^{2}) /27*9[/tex]
=[tex](100x^{5} y^{2} )/243[/tex]
Hence the product of expressions (5y/9)(x/3)(5y)(x^{3}/9)(4x) is
(100x^{5} y^{2} )/243.
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Answer:
5xy + 15y / 36x
Step-by-step explanation:
B.
Express with fractional exponents instead of radicals.
Answer:
I don't know
Step-by-step explanation:
980-788711758c75ds47c75c70rz76vyif7cuc8f58x85c7f85f8c
A triangle is dilated by a scale factor of n = One-third. Which statement is true regarding the dilation?
It is a reduction because n > 1.
It is a reduction because 0 < n < 1.
It is an enlargement because n > 1.
It is an enlargement because 0 > n > 1.
The true statement is (b) It is a reduction because 0 < n < 1.
How to determine the true statement?The scale factor of dilation is given as:
n = 1/3
When the scale factor (n) is between 0 and 1, then the dilation is considered reduction or compression;
Otherwise, the dilation is considered enlargement or stretch
Because n is between 0 and 1, then the dilation is a reduction
Hence, the true statement is (b) It is a reduction because 0 < n < 1.
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Answer:
b
Step-by-step explanation:
In the diagram below, if AB = 25, AC = 15, and BD = 5, then find the length of DC.
B
Select one:
a. 15
b. 5
c. 25
d. 3
D
PC
The length of DC based on the information given about the triangle is 19.37.
How to calculate the length?Based on the given information, it should be noted that:
AB² = AD² + BD²
25² = AD² + 5²
625 = AD² + 25
AD² = 625 - 25
AD² = 600
AD = √600
AD = 24.5
DC² = AD² + - AB² (Pythagoras theorem)
DC² = 24.5² + 15²
DC² = 375.25
DC = 19.37
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3 The ratio of hotdogs to hamburgers at the cookout 3:6. What is the difference between the number of hotdogs and hamburgers if there are 120 hamburgers?
The difference between the number of hotdogs and hamburgers is 60
How to determine the difference?The ratio is given as:
Hotdog : Hamburger = 3 : 6
When the total hamburger is 120.
We have:
Hotdog : 120 =3 : 6
Multiply 3 : 6 by 20
Hotdog : 120 =3 * 20 : 6 * 20
Evaluate
Hotdog : 120 = 60 : 120
By comparison, we have
Hotdog = 60
So, the difference is
Difference = 120 - 60
Evaluate
Difference = 60
Hence, the difference is 60
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a person is standing on top of a smaller building looking across the street at a taller building, which is 260 feet away horizontally. The angle of elavation from the top of the smaller building to the top of the building is 30 degrees and the angle of depression from the top of the smaller building to the base of the building is 20 degrees. how tall is the smaller building
The height of the smaller building is 94.63 feet, computed using the trigonometric ratios.
In the question, we take AB to be the taller building, where A is its top and B is its base, CD to be the shorter building, where C is its top and D is its base, and CE to be the perpendicular from C to AB.
Given that the horizontal distance between the two buildings is 260 feet, we can say that BC = CE = 260 feet.
The angle of elevation from the top of the shorter building to the top of the taller building is 30°, that is, ∠ECA = 30°.
The angle of depression from the top of the shorter building to the base of the taller building is 20°, that is, ∠ECB = 20°.
When ∠ECB = 20°, then ∠CBD = 20°, as they are alternate angles.
We are asked to find the height of the shorter building, that is, we are asked to find CD.
In ΔCBD,
tan ∠CBD = CD/BD {perpendicular/base},
or, tan 20° = CD/260,
or, CD = 260*tan 20° = 260*0.36397023426 {∵ tan 20° = 0.36397023426},
or, CD = 94.6322609092.
Therefore, the height of the smaller building is 94.63 feet, computed using the trigonometric ratios.
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The equivalent expression of 2a + 8 is ________.
Answer:
8+2aStep-by-step explanation:
Switching the order of the terms comes out as an equivalent expression.
Whatever A is, if we multiply it by 8 and add 2, it will be equivalent to the sum of the same number times multiplied by 8 and 2.
I hope that helps!
Answer:
2(a +4)
Step-by-step explanation:
Distributive property: a*b + a * c = a*(b+c)
Factorize 2 & 8 and find GCF of 2 & 8
2 = 2
8 = 2* 2*2
GCF = 2
2a + 8 = 2a + 2*4
= 2(a + 4)
If,0 < 3x - 2 < 1, what is one possible value for x?
Answer:
one possible solution is 5/6
Step-by-step explanation:
Solving for x
0 < 3x - 2 < 1
Add 2 to each side
0+2 < 3x - 2+2 < 1+2
2 < 3x<3
Divide each side by 3
2/3 < 3x/3 < 3/3
2/3 < x < 1
X must be between 2/3 and 1
Getting a common denominator
4/6 < x < 6/6
So x can be 5/6
In physics, a deceleration (the slowing down of an object in motion) is considered a negative acceleration (in other words, the opposite of speeding up). If you crash a small toy car into a wall, and the acceleration of the car is -3 m/s2, what would be the total acceleration you would have applied to the car, if you repeated the crash eleven times?
Using proportions, the total acceleration you would have applied to the car would be of -33 m/s².
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
For one crash, the acceleration is of -3 m/s². Hence, for 11 crashes, the total acceleration would be of:
a = -3 x 11 = -33 m/s².
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