By answering the above question, we may infer that Hence, when equation Mr. Haustra's tank is just 1/4 full, he can go around 70 miles.
What is equation?A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.
There are the following if Mr. Hamstra's automobile contains 14 2/5 gallons of fuel and his tank is only 1/4 full:
There are 3 1/2 gallons of gas in the tank (1/4) x 14 2/5.
If the automobile gets 20 miles per gallon on average, it may go as far as:
3 1/2 gallons x 20 miles per gallon is 70 miles.
Hence, when Mr. Hamstra's tank is just 1/4 full, he can go around 70 miles.
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If triangles QRS and LMN are congruent, then which two segments MUST be congruent?
All congruent segments are QR ≅ LM, RS ≅ MN and QS ≅ LN
What are congruent triangles?The triangles are said to be congruent if their corresponding sides and angles are congruent to each other.
Given are two congruent triangles QRS and LMN, we need to deduce the corresponding congruent parts,
So, according to the definition of congruent triangles :-
Sides :-
QR ≅ LM, RS ≅ MN and QS ≅ LN
Angles :-
∠ Q ≅ ∠ L, ∠ R ≅ ∠ M and ∠ S ≅ ∠ N
Hence, the congruent segments are QR ≅ LM, RS ≅ MN and QS ≅ LN
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Name Identities Idempotent laws A u A = A AnA = A Associative laws (AU B) U C = Au (B U C) (AnB) nc =An(Bnc) Commutative laws Au B = B U A AnB = BnA Distributive laws Au (B n C) = (Au B) n (Au C) An(B u C) (AnB) u (An C) Identity laws Au 0 = A AnU = A Domination laws Ano =0 Au U = U Double Complement law A = A AnA = U = 0 Au A = U 0 = U Complement laws De Morgans laws AuB = AnB AnB = Au B Absorption laws Au (AnB) = A An(A U B) = A
The Idempotent laws, Associative laws, Commutative laws, Distributive laws, Identity laws, Domination laws, Double Complement law, Complement laws, and De Morgan's laws are all types of identities that are used in set theory and Boolean algebra. These identities are used to simplify expressions and to prove the equivalence of two expressions.
The Idempotent laws state that the union or intersection of a set with itself is equal to the set itself:
A u A = A
AnA = A
The Associative laws state that the order in which sets are grouped does not matter when taking the union or intersection:
(AU B) U C = Au (B U C)
(AnB) nc =An(Bnc)
The Commutative laws state that the order in which sets are listed does not matter when taking the union or intersection:
Au B = B U A
AnB = BnA
The Distributive laws state that the union or intersection of a set with the intersection or union of two other sets is equal to the intersection or union of the set with each of the other sets:
Au (B n C) = (Au B) n (Au C)
An(B u C) (AnB) u (An C)
The Identity laws state that the union of a set with the empty set is equal to the set itself, and the intersection of a set with the universal set is equal to the set itself:
Au 0 = A
AnU = A
The Domination laws state that the intersection of a set with the empty set is equal to the empty set, and the union of a set with the universal set is equal to the universal set:
Ano =0
Au U = U
The Double Complement law states that the complement of the complement of a set is equal to the set itself:
A = A
AnA = U = 0
Au A = U 0 = U
The Complement laws state that the union of a set with its complement is equal to the universal set, and the intersection of a set with its complement is equal to the empty set:
AuB = AnB
AnB = Au B
De Morgan's laws state that the complement of the union of two sets is equal to the intersection of the complements of the sets, and the complement of the intersection of two sets is equal to the union of the complements of the sets:
AuB = AnB
AnB = Au B
The Absorption laws state that the union of a set with the intersection of the set with another set is equal to the set itself, and the intersection of a set with the union of the set with another set is equal to the set itself:
Au (AnB) = A
An(A U B) = A
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(1 point) Lelv1=−123−3,v2=16−4−4,v3=−1640Are the vectorsv1,v2andv3linearly independent? If the vectors are independent, enter zero in every answer blank since zeros are only the values that make the equation below true. If they are dependent, find numbers, not all zero, that make the equation below true. You should be able to explain and justify your answer.000=−123−3+16−4−4+−1640
v1, v2, and v3 are linearly independent
Yes, the vectors v1, v2, and v3 are linearly independent.
To verify this, we can set up the equation:
a1v1 + a2v2 + a3v3 = 0
If the vectors are linearly independent, then the only way to solve this equation is if a1 = a2 = a3 = 0.
Substituting the values for v1, v2, and v3 into the equation yields:
-123 - 3a1 + 16 - 4a2 - 4a3 - 1640a3 = 0
Solving for a1, a2, and a3, we get:
a1 = 0, a2 = 0, a3 = 0
Therefore, the vectors v1, v2, and v3 are linearly independent.
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I WILL GIVE YOU BRAINLIEST IF YOU ANSWER THIS!
Identify the asymptotes of the rational function.
[tex]y = \frac{1}{3x + 9} - 9[/tex]
A. Vertical asymptote: x = −3
Horizontal asymptote: y = –9
B. Vertical asymptote: x = –9
Horizontal asymptote: y = −3
C. Vertical asymptote: x = 0
Horizontal asymptote: y = 9
D. Vertical asymptote: x = 3
Horizontal asymptote: y = 9
Answer:
A) vertical asymptote: X = -3
Horizontal asymptote: y = -9
Step-by-step explanation:
sorry I don't know how to explain it
Sandy decided to do an experiment with a sandwich she left in her locker. She found it 4 days after she had put it there, took it to the lab, and counted 71 bacteria. 3 days later she counted 185 bacteria.
Write the exponential equation that represents this problem.
She estimates that it will take 500 bacteria to completely cover her sandwich. How many days would it take this to happen, starting from the day she first put her sandwich in her locker?
The respοnse tο the given questiοn wοuld be that As a result, it will take equatiοn arοund 9.7 days frοm the day Sandy placed her sandwich in her lοcker fοr 500 bacteria tο cοmpletely cοver it.
What is equatiοn?When twο statements are cοnnected by a mathematical equatiοn, the equals sign (=) implies equality. An equatiοn in algebra is a mathematical statement that prοves the equivalence οf twο mathematical expressiοns. Fοr instance, the equal sign separates the numbers in the equatiοn 3x + 5 = 14. It is pοssible tο determine the relatiοnship between the twο sentences οn either side οf a letter using a mathematical fοrmula. The lοgο fοr the particular piece οf sοftware is frequently the same. as 2x - 4 = 2, fοr dinstance.
This issue's expοnential equatiοn is as fοllοws:
N(t) = N0 × [tex]e^{(kt)[/tex] × (kt)
We can utilise the abοve infοrmatiοn tο determine the values οf N0 and k:
N(4) = 71 × N(7) = 185
When we enter these numbers intο the equatiοn, we οbtain:
71 = N0 × [tex]e^{(4k)[/tex] × (4k)
185 = N0 × [tex]e^{(7k)[/tex] × (7k)
185/71 = [tex]e^{(3k)[/tex] × (3k)
As a result, the equatiοn fοr this issue is:
N(t) = 23.77 × [tex]e^{(0.3056t)}[/tex] × (0.3056t)
We may set N(t) equal tο 500 and sοlve fοr t tο see hοw many days it will take fοr 500 germs tο grοw:
500 = 23.77 × [tex]e^{(0.3056t)}[/tex] × (0.3056t)
9.7 days based οn t = ln(500/23.77) / 0.3056.
As a result, it will take arοund 9.7 days frοm the day Sandy placed her sandwich in her lοcker fοr 500 bacteria tο cοmpletely cοver it.
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condense the expression log3 2x+3 log3 4x
Answer:
The expression log3 2x + 3 log3 4x can be condensed to log3 8x + 3. This is because log3 4x = 2 log3 2x, so when we add the two logarithms together, the result is log3 8x + 3
Step-by-step explanation:
A pair of shoes is on sale for 35% off and regularly cost $70. If sales tax is 5%, what is the amount you will pay for the shoes?
Answer:
The discounted price of the shoes is:
$70 x 0.65 = $45.50
The amount of sales tax is:
$45.50 x 0.05 = $2.275
Therefore, the total amount you will pay for the shoes is:
$45.50 + $2.275 = $47.775
Rounded to two decimal places, the amount you will pay is $47.78.
The length of a rectangle is 2 inches more than its width. Write an equation relating the length l of the rectangle to its width w
Answer: If the length of a rectangle is 2 inches more than its width, we can write:
length = width + 2
We can also rearrange this equation to solve for the width in terms of the length:
width = length - 2
Both of these equations relate the length l of the rectangle to its width w.
Step-by-step explanation:
1. After watching "Nike an Example of a TNC (Transnational Corporation)" and reviewing your readings and notes, what is the effect of Nike's production of shoes and clothing around the world has on the standard of living of workers in the countries where is produces ("the global South") and in the countries where it sells its products ("the global North")?
After watching "Nike an Example of a TNC (Transnational Corporation)" and reviewing the readings and notes, it is clear that Nike's production of shoes and clothing around the world has had a significant effect on the standard of living of workers in the countries where it produces ("the global South") and in the countries where it sells its products ("the global North").
In the global South, Nike's production has led to the creation of jobs and an increase in income for workers. However, these workers often face poor working conditions and low wages, which can negatively impact their standard of living. Additionally, the environmental impact of Nike's production can also have a negative effect on the standard of living for workers and their communities.
In the global North, Nike's production has led to the availability of affordable and fashionable clothing and shoes. However, the low prices of these products are often achieved through the exploitation of workers in the global South, which can lead to ethical concerns for consumers. Additionally, the environmental impact of Nike's production can also have a negative effect on the standard of living for consumers and their communities.
Overall, Nike's production of shoes and clothing around the world has had a significant effect on the standard of living of workers in the global South and consumers in the global North. While there are some benefits, such as job creation and affordable products, there are also significant negative impacts, including poor working conditions, low wages, and environmental degradation.
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The expression below shows how to convert 12 gallons to cups with one part missing 4 cups 1 quart Which value completes the expression? 12 gallons 1 4 gallons 1 quart 1 gallons 4 quarts 4 quarts 1 gallon 16 cups 1 gallon
Answer: 4 quarts / 1 gallon
Step-by-step explanation: or C!! hope this helps!!
Help me find the g(x)
This is exponential function
The function g(x) is given as follows:
g(x) = -7x + 15What is the value of g(x) if the exponential function f(x) = -7ˣ⁺¹ shifts two units to the right?To shift the exponential function f(x) = -7x+1 two units to the right, we need to replace x with (x-2) in the equation.
This is because when x is replaced with (x-2), the function will have the same output value as before when x was 2 units less.
Thus, the shifted function g(x) can be written as:
g(x) = f(x-2)
g(x) = -7(x-2) + 1
g(x) = -7x + 15
So, the function g(x) is given by g(x) = -7x + 15, and it is the same as f(x) shifted 2 units to the right.
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What is the properties of.
|+-8|-5=7
The expression after applying distributive property is
[(-5/8) × (-3)] + [(-5/8 ) × (-7)] = 50/8
What is Distributive Property?Distributive property shows us how to multiply an arithmetic number or an algebraic expression with two or more numbers or expressions within brackets.
This can be symbolically represented as :
X ( Y + Z ) = XY + XZ
here, we have,
The expression given here is -5/8 (-3 + (-7)).
Applying distributive property, we get,
-5/8 (-3 + (-7)) = [(-5/8) × (-3)] + [(-5/8 ) × (-7)]
= (15/8) + (35/8)
= 50/8
The expression - 5/8(-3 + (-7)) after applying distributive property changes to the expression [(-5/8) × (-3)] + [(-5/8 ) × (-7)] = 50/8.
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You got home on day two in 75% of the time it took you on day one. On day three it took you 20% longer than on day two. On day four it took you two minutes less than on day three. On day five you walked 50% faster than on day four and it took you 15 minutes to get home. How long did it take you to get home on day one?
So the answer is: it took 35.56 minutes to get home on day one.
To find out how long it took to get home on day one, we need to work backwards from day five. Here's how:
On day five, it took 15 minutes to get home. This was 50% faster than on day four, which means that on day four it took 15 minutes / 0.5 = 30 minutes.
On day four, it took 30 minutes, and this was 2 minutes less than on day three. So on day three it took 30 minutes + 2 minutes = 32 minutes.
On day three, it took 32 minutes, and this was 20% longer than on day two. So on day two it took 32 minutes / 1.2 = 26.67 minutes.
On day two, it took 26.67 minutes, and this was 75% of the time it took on day one. So on day one it took 26.67 minutes / 0.75 = 35.56 minutes.
So the answer is: it took 35.56 minutes to get home on day one.
Here's the answer formatted with HTML:
To find out how long it took to get home on day one, we need to work backwards from day five. Here's how:
So the answer is: it took 35.56 minutes to get home on day one.
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[tex]G(3)=4^{2}[/tex]
Answer:
G = 5.333
Step-by-step explanation:
Given: G(3) = 4^2
3 times G is 3G:
3G = 4^2
4 squared is 16:
3G = 16
Divide both sides by 3:
G = 5.333
How do I solve this?
The value of x is 4 using the trigonometric function.
What is hypotenuse?The term "hypotenuse" refers to a right-angled triangle's longest side as compared to the base and perpendicular lengths. The right angle, the largest angle of the three angles in a right triangle, lies opposite the hypotenuse side. In essence, only the right triangle possesses the hypotenuse feature, not any other triangles. When we study the right-angled theorem, Pythagoras Theorem, or Pythagorean Theorem, this is better understood. The main use of these ideas is in trigonometry.
The triangle is a right-triangle.
Thus, using the trigonometric functions we have:
Sin (30) = opposite / hypotenuse
sin (30) = 2 / x
1/2 = 2/x
x =2(2)
x = 4
Hence, the value of x is 4 using the trigonometric function.
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Hence determine the circumference of the base of the tin In mm
The required real diameter οf the tin is 250 mm.
What is the diameter?Diameter is a measurement οf the length οf a straight line that passes thrοugh the center οf a circular οbject, such as a circle οr a sphere. It is the distance between twο pοints οn the οbject's edge that are furthest frοm each οther and that pass thrοugh the center οf the οbject.
Here,
If the diameter οf the tin in the picture is 10 cm, then the actual diameter οf the tin can be calculated as:
Actual Diameter = 2.5 × Picture Diameter
Actual Diameter = 2.5 × 10 cm
Actual Diameter = 25 cm
Tο cοnvert the actual diameter tο millimeters (mm), we can multiply by 10:
Actual Diameter in mm = Actual Diameter × 10
Actual Diameter in mm = 25 cm × 10
Actual Diameter in mm = 250 mm
Therefοre, the real diameter οf the tin is 250 mm.
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Complete question:
IM DESPERATE PLS HELP PLEASE
The area of the one rectangular panel is equal to 220 ft².
How to calculate the area of a rectangle?Mathematically, the area of a rectangle can be calculated by using this formula:
A = LW
Where:
A represents the area of a rectangle.W represents the width or base of a rectangle.L represents the length or height of a rectangle.Generally speaking, the opposites sides of a rectangle are equal;
DN = WF
12x + 4 = 20x - 8
20x - 12x = 8 + 4
8x = 12
x = 3/2.
Side DN = 12x + 4 = 12(3/2) + 4 = 22 feet.
DW = NF
5y - 5 = 2y + 4
y = 3
Side DW = 5y - 5 = 5(3) - 5 = 10 feet.
Area of rectangular panel = DN × DW
Area of rectangular panel = 22 × 10
Area of rectangular panel = 220 ft².
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Mr. Smith is putting a brick border around his irregular shaped yard. Before installing the
border, he must cut the bricks to fit the angles of the garden. Use the given measures to answer
the question.
∡ = (7 − 8)°
∡ = (4 + 46)°
∡ = (5)°
∡ = (6 + 12)°
∡ = ( + 7)°
∡ = (5 − 9)°
What are the angle measures of each vertex of the yard? Show your work
All the angle measures are as follows: m∠A = 160°, m∠B = 142°, m∠C = 120°, m∠D = 156°, m∠E = 31°, m∠F = 111°
We know that a hexagon is a six-sided polygon and also a convex hexagon which basically means that the hexagon's vertices are pointed outwards.
And with no interior angle has an angle measuring of more than 180°.
We know that the number of sides in the polygon = 6.
And the total sum of the irregular angle = (4 - 2) x 180° = 720°.
7x - 8 + 4x + 46 + 5x + 6x + 12 + x + 7 + 5x - 9 = 720°.
28x = 720 - 48
x = 24
Substituting all the values of x = 24 to all the interior angle measures,
Now, all the measures are:
m∠A = 7x - 8 = 160°
m∠B = 4x + 46 = 142°
m∠C = 5x = 120°
m∠D = 6x + 12 = 156°
m∠E = x + 7 = 31°
m∠F = 5x - 9 = 111°
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A= (5,-5) B=(9,9)
The point on the line segment connecting A and B that
is 3/4 of the way from A to B is: (? ,? )
The point on the line segment that is 3/4 of the way from A to B is (8,5.5).
To find the point on the line segment that is 3/4 of the way from A to B, we can use the formula for finding a point on a line segment:
P = (1-t)A + tB
Where P is the point we are trying to find, A and B are the endpoints of the line segment, and t is the fraction of the distance from A to B that we want to find. In this case, t = 3/4.
Plugging in the values for A, B, and t, we get:
P = (1-3/4)(5,-5) + (3/4)(9,9)
Simplifying, we get:
P = (1/4)(5,-5) + (3/4)(9,9)
P = (1.25,-1.25) + (6.75,6.75)
P = (8,5.5)
So the point on the line segment that is 3/4 of the way from A to B is (8,5.5).
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The table below shows the ages of 50 people on a bus.
Age (years) Frequency
0 < h ≤ 20 20
20 < h ≤ 40 10
40 < h ≤ 60 10
60 < h ≤ 80 10
Calculate an estimate for the mean age.
An estimate for the mean age of the 50 people on the bus is 34 years.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
Let us calculate the sum of the products of the midpoints and their respective frequencies, and divide by the total frequency to obtain the estimated mean age.
The midpoint and frequency data for each age group can be represented as follows:
Age group Midpoint Frequency
0 < h ≤ 20 10 20
20 < h ≤ 40 30 10
40 < h ≤ 60 50 10
60 < h ≤ 80 70 10
To calculate the estimated mean age, we can use the formula:
Estimated mean age = (Σ(midpoint × frequency)) / total frequency
Estimated mean age = ((10 × 20) + (30 × 10) + (50 × 10) + (70 × 10)) / (20 + 10 + 10 + 10)
Estimated mean age = (200 + 300 + 500 + 700) / 50
= 1700 / 50
= 34
Therefore, an estimate for the mean age of the 50 people on the bus is 34 years.
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NEED HELP FAST! 20 POINTS!!
The value of x is 57 degrees. This was obtained by the use of the trigonometric ratios.
What is a right angle triangle?A right triangle, also known as a right-angled triangle, is one that we can say that an angle in this triangle would be 90 degrees.
The Pythagorean theorem can be used to obtain the lengths of the various sides that we have in the right angled triangle. This is one of the fundamental theories in the right angled triangle.
Let us now use the cosine to find the angle as shown below;
We know that;
Cos x = 11/20
x = Cos-1(11/20)
x = 57 degrees
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Select the correct answer. Which inequality represents all the solutions of 10(3x + 2) > 7(2x − 4)?
Answer:
x > -3
Step-by-step explanation:
10(3x + 2) > 7(2x − 4)
30x + 20 > 14x - 28
16x + 20 > -28
16x > -48
x > -3
So, the answer is x > -3
Use long division to find the quotient
(2x^3-5x^2+3x-6) divided by (x+4)
Using long division the quotient of (2x^3-5x^2+3x-6) divided by (x+4) is 2x^2 - 13x + 55.
What is quotient?Quotients are often used in algebra and other branches of mathematics to solve equations and expressions involving division. They are also commonly used in everyday life, such as when calculating the cost per unit of a product, or the average speed of a journey.
For example, if we divide 10 by 2, the quotient would be 5, since 10 divided by 2 equals 5. Similarly, if we divide 12 by 3, the quotient would be 4, since 12 divided by 3 equals 4.
Now let find the quotient using long division:
2x^2 - 13x + 55
_______________________
x + 4 | 2x^3 - 5x^2 + 3x - 6
- (2x^3 + 8x^2)
---------------
- 13x^2 + 3x
- (-13x^2 - 52x)
---------------
55x - 6
- (55x + 220)
-----------
-226
Therefore, the quotient is 2x^2 - 13x + 55 and the remainder is -226.
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1. What is the product of (5x - 2) and (6x2 +7x - 4)? Write your answer in standard form.
(a) Show your work.
(b) Is the product of (5x - 2) and (6x2 +7x - 4) equal to the product of (2 – 5x) and (6x2
+7x - 4)? Explain your answer.
a. The product of (5x-2) and (6x²+7x -4 ) is 30x³+23x²-34x+8
b. The product of (5x - 2) and (6x2 +7x - 4) is not equal to The product of (2 – 5x) and (6x2
+7x - 4)
What is product?Product is results that comes from multiply two or more terms or expression together.
For example, 5x × 2 = 10. The product here is 10 and the two terms are 5x and 2.
therefore,
(5x-2)( 6x²+7x-4)
using distributive law
5x( 6x²+7x-4) + (-2)(6x²+7x-4)
= 30x³+ 35x²-20x -12x²-14x+8
collect like terms
30x³ + 35x²-12x²-20x-14x+8
= 30x³+23x²-34x+8
b. The product of (2 – 5x) and (6x2 +7x - 4)
= 2(6x2+7x - 4) -5x(6x2+7x - 4)
= 12x²+14x-8-30x³-35x²-20x
= -30x³+12x²-35x²+14x-20x-8
= -30x³ - 23x²-6x-8
therefore the product of (5x - 2) and (6x2 +7x - 4) is not equal to The product of (2 – 5x) and (6x2
+7x - 4)
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What 's the area which has sides of length 6and8
Answer: 48
Step-by-step explanation:
6 x 8 = 48
provealgebraically that the square of any odd number is always 1more than a multiple of 8
Answer:
Let's assume that we have an odd integer represented by the variable "n". Then, according to the definition of an odd integer, we can write it as follows:
n = 2k + 1
Where "k" is any integer.
Now, we need to find the square of "n". We can write it as follows:
n^2 = (2k + 1)^2
n^2 = 4k^2 + 4k + 1
Now, we can simplify the expression:
n^2 = 4k(k + 1) + 1
We can observe that the expression 4k(k + 1) is always an even number. This is because when we multiply two consecutive integers (k and k + 1), we always get an even result. Therefore, we can write:
4k(k + 1) = 8m
Where "m" is any integer. Substituting this expression into the previous equation, we get:
n^2 = 8m + 1
This shows that the square of any odd number is always 1 more than a multiple of 8, which is what we wanted to prove.
Answer:
The answer is 8
Step-by-step explanation:
If n is "any integer", then 2n+1 is "any odd number."
The square of any odd number is then ...
(2n+1)² = 4n² +4n +1 = 4n(n+1) +1
Since n is any integer, one of n and n+1 will be an even integer, so the product 4n(n+1) will be divisible by 8.
Then the sum 4n(n+1) +1 is one more than a number divisible by 8, hence ...
the square of an odd number is 1 more than a multiple of 8.
Hi due today pls help! Ty! question attached
Snails to Newts iz
There are 36 Snails present in the pond.
What is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
Given:
Ratio of Snail to Newts = 3:4
So, Number of Snail= 3x and number of Newts = 4x
If 12 more newts added then the ratio is 3:5
So, 3x/ 4x+ 12 = 3/5
15 x = 3(4x+ 12)
15x = 12x + 36
15x - 12x = 36
3x = 36
x= 12
So, the number of Snails in pond are 3x = 3(12)= 36.
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The map shows the distance in air miles from Houston to both Austin and San Antonio.What is the greatest possible distance to Austin to san Antonio
a. The greatest possible distance from Austin to san Antonio is 335.77 mi, b. the Least possible distance to Austin to san Antonio is 120.03 mi.
Describe Distance?Distance refers to the physical length or space between two points or objects. It is an important concept in mathematics, physics, and everyday life. Distance can be measured in various units such as meters, feet, kilometers, miles, or light-years, depending on the context and scale of the measurement.
In mathematics, distance is often measured using the Euclidean distance formula, which calculates the distance between two points in a plane or in three-dimensional space. The formula is given by:
d = √((x2 - x1)² + (y2 - y1)²)
where d is the distance between the two points, (x1, y1) and (x2, y2).
In physics, distance is often used to describe the separation between two objects or particles. The distance between two objects can affect the strength of gravitational or electromagnetic forces between them. Distance is also an important concept in special relativity, where it is affected by the relative motion of observers and the distortion of spacetime.
a. Greatest possible distance from Austin to san Antonio= Distance from Austin →Houston → san Antonio.
Austin - Houston = 146.43 mi
Houston - San Antonio = 189.34 mi
Total distance= 146.43+ 189.34= 335.77 mi
b. Least possible distance to Austin to San Antonio = the shortest distance = x
By Pythagoras theorem
(189.34)² = (146.43)²+ x²
x² = (189.34)²- (146.43)²
x² = 35849.63- 21441.744
x² = 14407.89
x =√14407.89
x = 120.03
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The complete question is:
Function r is an input-output machine that uses the rule "3(t^4)-19". Write the machine's function using function notation.
The given function 3(t^4)-19 is thus written using the function rule in notations as: f(t) = 3(t^4)-19 for the input-output machine.
Explain about the function notation?Although equations are frequently used to represent functions, a sequence of equations all beginning with y = may make it difficult to distinguish between different functions.
So, giving functions names is a great touch. Functions and their variables are named in function notation.A function that involves the variables x and y can be expressed in the form y = "some expression involving x," or y = f(x) .By reading the last sentence as "y equals f of x," it is indicated because y is a function of x.A function called f is defined by the notation f(x).
This should be understood as "y is a function of x."The input value, called independent variable, is represented by the letter x.The output value, meaning dependent variable, is represented by f(x) in place of the letter y.The given function 3(t^4)-19 is thus written using the function rule in notations as:
f(t) = 3(t^4)-19.
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1.In a circle of radius 6cm a chord is drawn 3cm from the center
a) Calculate the angle subtended by the chord at the centre of the circle b)Find the length of the minor and the perimeter of the minor segment formed by a chord and minor
Answer:
a) The angle subtended by the chord at the centre of the circle is 120°.
b) The length of the minor arc is 12.57 cm (2 d.p.).
The perimeter of the minor segment formed by a chord and minor arc is 22.96 cm (2 d.p.).
Step-by-step explanation:
The chord of a circle is the base of an isosceles triangle, where its congruent sides are the radii of the circle.
The height of the isosceles triangle is the perpendicular bisector of the chord from the central angle.
The angle subtended by the chord at the centre of the circle is twice the angle formed by the radius and the perpendicular bisector (marked θ on the attached diagram).
To find the measure of angle θ, use the cosine trigonometric ratio.
[tex]\begin{aligned}\implies \cos \theta&=\sf \dfrac{adjacent\;side}{hypotenuse}\\\\\cos \theta &=\sf \dfrac{3}{6}\\\\\cos \theta &=\sf \dfrac{1}{2}\\\\\theta &=\arccos\left(\sf \dfrac{1}{2}\right)\\\\\theta &=60^{\circ}\end{aligned}[/tex]
As the angle subtended by the chord at the centre of the circle is twice angle θ:
[tex]\implies 2\theta=2 \cdot 60^{\circ}=120^{\circ}[/tex]
Therefore, the angle subtended by the chord at the centre of the circle is 120°.
A minor arc is less than 180° and is equal to the central angle.
Therefore, as the angle subtended by the chord at the centre of the circle is 120°, the minor arc is the part of the circumference between the two endpoints of the chord (marked in red on the attached diagram).
To find the length of the minor arc, use the arc length formula.
[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Arc length}\\\\Arc length $= \dfrac{\pi r\theta}{180^{\circ}}$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in degrees.\\\end{minipage}}[/tex]
Therefore:
[tex]\begin{aligned}\implies \textsf{Arc length}&= \dfrac{\pi \cdot 6 \cdot 120^{\circ}}{180^{\circ}}\\\\&= 4 \pi\\\\&= 12.57\; \sf cm\;(2\;d.p.)\end{aligned}[/tex]
Therefore, the length of the minor arc is 12.57 cm (2 d.p.).
To calculate the perimeter of the minor segment formed by the chord and minor arc, we need to calculate the length of the chord.
[tex]\boxed{\begin{minipage}{10.2 cm}\underline{Chord length}\\\\Chord length $=2r\sin\left(\dfrac{\theta}{2}\right)$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle (in degrees) subtended at the center by the chord.\\\end{minipage}}[/tex]
Therefore:
[tex]\begin{aligned}\implies \textsf{Chord length}&= 2 \cdot 6 \cdot \sin \left(\dfrac{120^{\circ}}{2}\right)\\\\&= 12 \cdot \sin \left(60^{\circ}\right)\\\\&= 12 \cdot \dfrac{\sqrt{3}}{2}\\\\&= 6\sqrt{3}\; \sf cm\end{aligned}[/tex]
Therefore, the perimeter of the minor segment formed by the chord and minor arc is:
[tex]\begin{aligned}\implies \textsf{Perimeter}&=\textsf{Minor arc length}+\textsf{Chord length}\\&=4 \pi + 6 \sqrt{3}\\&=22.96\; \sf cm\;(2\;d.p.)\end{aligned}[/tex]