So the slope-intercept form of the equation of the line passing through the two given points is:
y = -5x + 26
What distinguishes a linear equation?If an equation is expressed in the form y=mx+b, where m denotes the slope and b the y-intercept, it is said to be linear.
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
To find the slope of the line passing through the two given points (5,1) and (2,16), we can use the formula:
m = (y2 - y1)/(x2 - x1)
where (x1, y1) = (5,1) and (x2, y2) = (2,16)
m = (16 - 1)/(2 - 5) = -5
So the slope of the line is -5.
To find the y-intercept b, we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
Using the point (5,1) and the slope m = -5, we get:
y - 1 = -5(x - 5)
Simplifying this equation, we get:
y = -5x + 26
So the slope-intercept form of the equation of the line passing through the two given points is:
y = -5x + 26
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The area of a rectangular tablecloth is 8 3/4 square yards. The width of the tablecloth is 1 7/8 yards. What is the length, in yards, of the tablecloth?
The width of the rectangle is 4 2/3 yards.
Area of the rectangle:The area of a rectangle is the measure of the total region that the rectangle occupies in two-dimensional space.
It is calculated by multiplying the length of the rectangle by its width.
The formula for the area of a rectangle is:
Area = length × widthHere we have
The area of a rectangular tablecloth is 8 3/4 square yards.
The width of the tablecloth is 1 7/8 yards
For simple calculation convert given fractions into improper fractions
=> 8 3/4 = 35/4 yards
=> 1 7/8 = 15/8 yards
Let 'x' be the breadth of the rectangle
As we know Area of the rectangle = Length × width
=> x × 15/8 = 35/4
=> x = 35/4 × 8/15
=> x = 14/3
Here, 14/3 = 4 2/3 yards
Therefore,
The width of the rectangle is 4 3/2 yards.
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Which table shows a proportional relationship between x and y?
x 1 2 3 4
y 6 8 10 12
x 4 7 8 10
y 2 3.5 4 5
x 40 50 60 90
y 8 10 14 18
x 2 4 7 8
y 6 10 21 24
Answer:
Second option
Step-by-step explanation:
Check the relationship between the x digits and compare with y digits. It must be the same all through
In the second option x =2y for all the digits
4:2, 7:3.5,8:4,10:5
5 boys and girls are running in a marathon. How many ways can the first finishen complete the marathon it: a) the first 2 finishers must have different genders? b) Chantal must finish the race before David?
There are 8 possible ways for the first 2 finishers to complete the marathon with different genders. There are 20 possible ways for Chantal to finish the race before David.
There are a couple of different ways to approach this problem, but one common method is to use the multiplication principle, which states that if there are A ways to do one thing and B ways to do another, then there are A*B ways to do both. We can apply this principle to both parts of the question.
a) If the first 2 finishers must have different genders, then we can think about the possible combinations of boys and girls. There are 2 options for the first finisher (either a boy or a girl), and then there are 4 options for the second finisher (either a boy or a girl, but not the same gender as the first finisher).
So, using the multiplication principle, we can find the total number of ways for the first 2 finishers to complete the marathon with different genders:
2 * 4 = 8
Therefore, there are 8 possible ways for the first 2 finishers to complete the marathon with different genders.
b) If Chantal must finish the race before David, then we can think about the possible positions for Chantal and David. There are 5 possible positions for Chantal (first, second, third, fourth, or fifth), and then there are 4 possible positions for David (second, third, fourth, or fifth, but not before Chantal).
So, using the multiplication principle, we can find the total number of ways for Chantal to finish before David:
5 * 4 = 20
Therefore, there are 20 possible ways for Chantal to finish the race before David.
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The permutation ([1,2,3,4,5,6],[6,5,4,3,1,2]) can be written as a product of disjoint cycles as (A) ([1,6,2,5])([3,4]) (B) ([1,2,3,4])([5,6]) (C) ([1,3,4,5])([2,6]) (D) ([1,2,5,6])([3,4])
The permutation ([1,2,3,4,5,6],[6,5,4,3,1,2]) can be written as a product of disjoint cycles as (A) ([1,6,2,5])([3,4]).
To find the product of disjoint cycles, we can start by looking at the first element in the first cycle, which is 1. We see that 1 maps to 6 in the given permutation, so we write ([1,6. Next, we see that 6 maps to 2, so we write ([1,6,2. Finally, 2 maps to 5, so we write ([1,6,2,5. Since 5 maps back to 1, we can close the cycle and write ([1,6,2,5]).
Next, we look at the remaining elements, which are 3 and 4. We see that 3 maps to 4 and 4 maps back to 3, so we write ([3,4]).
Therefore, the permutation ([1,2,3,4,5,6],[6,5,4,3,1,2]) can be written as a product of disjoint cycles as ([1,6,2,5])([3,4]). The correct answer is (A) ([1,6,2,5])([3,4]).
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PLEASE HELP
A cylinder-shaped container is used to store water. The container has a height of 6 feet and
diameter of 3 feet.
About how much water is in the container when it is 3/4 full?
o 127 cubic feet
o 42 cubic feet
o 32 cubic feet
o 14 cubic feet
Answer:
32 cubic feet
Step-by-step explanation:
The formula for a cylinder is [tex]\pi r^{2} h[/tex].
The radius of the cylinder is equal to 1.5 feet, since it is [tex]\frac{diameter}{2}[/tex].
Plugging in, the cylinder's full volume is [tex]6\pi 1.5^2[/tex] which is approximately 42.4 cubic feet.
To find the amount of water when it is 3/4 full, multiply 42.4 x .75, to get around 31.8, and 32 when rounded
9 of 119 of 11 Items
Question
Jillian and Jacob are playing a game where they have to collect blue and yellow tokens. Blue and yellow tokens are worth a different amount of points. Jillian has collected 4 blue tokens and 7 yellow tokens and has 95 points. Jacob has collected 4 blue tokens and 3 yellow tokens and has 75 points. How many points are a blue token and a yellow token worth?
Question
Jillian and Jacob are playing a game where they have to collect blue and yellow tokens. Blue and yellow tokens are worth a different amount of points. Jillian has collected 4 blue tokens and 7 yellow tokens and has 95 points. Jacob has collected 4 blue tokens and 3 yellow tokens and has 75 points. How many points are a blue token and a yellow token worth?
A blue token is worth 15 points and a yellow token is worth 5 points.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Let b represent the point value of a blue token and
y represent the point value of a yellow token.
We can set up two equations:
4b + 7y = 95 (Jillian's score)
4b + 3y = 75 (Jacob's score)
Subtracting the second equation from the first, we get:
4b + 7y - (4b + 3y) = 95 - 75
Simplifying this, we get:
4y = 20
Dividing both sides by 4, we get:
y = 5
So a yellow token is worth 5 points.
Now we can substitute this value of "y" into one of the original equations and solve for "b".
4b + 7y = 95
4b + 7(5) = 95
4b + 35 = 95
4b = 60
b = 15
So a blue token is worth 15 points.
Therefore, a blue token is worth 15 points and a yellow token is worth 5 points.
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Simplify completely. Assume no denominators are zero. 2x^3 – 2x^2 – 40x/2x^2+5x-12÷ 6x^4 – 150x^2/8x^3 - 27
Simplifying the expression (2x³ – 2x² – 40x )/(2x² + 5x - 12) ÷ (6x⁴ – 150x²)/(8x³ - 27) will result to (x - 5)/(3x(x - 3)(x + 5)).
To simplify the given expression completely, we need to factor the numerator and denominator of each fraction, and then simplify by canceling out common factors.
(2x³ – 2x² – 40x )/(2x² + 5x - 12) ÷ (6x⁴ – 150x²)/(8x³ - 27)
= (2x(x² - x - 20))/(2x^2 + 5x - 12) ÷ (6x^2(x^2 - 25))/(8x^3 - 27)
= (2x(x - 5)(x + 4))/(2x² + 8x - 3x - 12) ÷ (6x²(x - 5)(x + 5))/(8x³ - 27)
= (2x(x - 5)(x + 4))/(2x(x + 4)(x - 3)) ÷ (6x²(x - 5)(x + 5))/(8x³ - 27)
= (2x(x - 5)(x + 4))/(2x(x + 4)(x - 3)) × (8x³ - 27)/(6x²(x - 5)(x + 5))
= (x - 5)/(x - 3) × (8x³ - 27)/(6x²(x - 5)(x + 5))
= (x - 5)/(x - 3) × (2x - 3)(4x² + 6x + 9)/(6x²(x - 5)(x + 5))
= (x - 5)/(x - 3) × (2x - 3)/(6x(x - 5)(x + 5))
= (x - 5)/(3x(x - 3)(x + 5))
Therefore, the simplified expression is (x - 5)/(3x(x - 3)(x + 5)).
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A (-1,6)
Work out the length of AB.
Give your answer to 3 significant figure
O
B (5, 3)
thagoras' Theorem - Line on a Graph
8
X
Answer:
AB = 6.71
Step-by-step explanation:
The vertical leg of the right triangle
= absolute value of difference in y-coordinates between A(-1, 6) and B(5, 3)
= |6 - 3| = |3|
= 3
The horizontal leg of the right triangle
= absolute value of difference in x-coordinates between A(-1, 6) and B(5, 3)
= |- 1 - 5|
=|- 6|
= 6
By the Pythagorean theorem, the hypotenuse AB is related to each of these two legs by the formula
AB² = 3² + 6²
AB² = 9 + 36
AB² = 45
AB = √45
or
AB = 6.7082039324
= 6.71 significant to 3 significant figures
Significant figures means number of digits excluding leading and trailing zeros
Question 13 Evaluate the limit: lim−(x−>9)(7x−63)/(x^2−2x−63)=
Use the result of Example 1.3 to verify the formulas it Prove that x+y divides x^2n+1+y^2n+1 for n≥0. Cher then n is a positive integer, so also is (n^3
The limit of the given function can be evaluated by simplifying the expression and then substituting the value of x.
First, we can factor the numerator and denominator of the expression:
lim−(x−>9)(7x−63)/(x^2−2x−63)=lim−(x−>9)(7(x−9))/((x−9)(x+7))
Next, we can cancel out the common factor of (x-9) from the numerator and denominator:
lim−(x−>9)(7)/(x+7)
Now, we can substitute the value of x=9 into the expression:
lim−(x−>9)(7)/(x+7)=7/(9+7)=7/16
Therefore, the limit of the given function is 7/16.
As for the second part of the question, we can use the result of Example 1.3 to verify the formulas. According to Example 1.3, if x+y divides x^n+y^n for n≥0, then x+y also divides x^(n+1)+y^(n+1). Using this result, we can prove that x+y divides x^2n+1+y^2n+1 for n≥0 by induction.
Base case: n=0
x^2(0)+1+y^2(0)+1=x+y, which is divisible by x+y.
Inductive step: Assume that x+y divides x^2n+1+y^2n+1 for some n≥0. We need to show that x+y divides x^2(n+1)+1+y^2(n+1)+1.
x^2(n+1)+1+y^2(n+1)+1=(x^2n+1+y^2n+1)(x^2+y^2)
Since x+y divides x^2n+1+y^2n+1 by assumption, and x+y divides x^2+y^2 by Example 1.3, x+y also divides (x^2n+1+y^2n+1)(x^2+y^2) by the distributive property. Therefore, x+y divides x^2(n+1)+1+y^2(n+1)+1 for n≥0.
Finally, for the third part of the question, we can use the fact that n is a positive integer to show that (n^3) is also a positive integer. Since n is a positive integer, n^3 is also a positive integer because the product of three positive integers is a positive integer. Therefore, (n^3) is a positive integer.
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A manufacturer of electronic calculators takes a random sample of 1200 calculators and finds 5 defective units. Construct a 95% confidence interval on the population proportion.
(Round the answers to 4 decimal places.)
(_____< p <_______)
The 95% confidence interval for the population proportion of defective calculators is (0.0015, 0.0069).
Hence, (0.0015 < p < 0.0069).
According to the given information
we can use the formula for calculating the confidence interval for a population proportion,
⇒ P ± z√(P(1-P )/n)
Where,
P = sample proportion of defective calculators = 5/1200
= 0.0042
n = sample size = 1200
z = z-score for 95% confidence level = 1.96 (from standard normal distribution table)
Substituting the values in the formula, we get,
⇒ 0.0042 ± 1.96√(0.0042(1-0.0042)/1200)
Simplifying the expression gives us the confidence interval,
⇒ (0.0015 < p < 0.0069)
Therefore,
The 95% confidence interval for the population proportion of defective calculators is (0.0015, 0.0069).
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kierna is starting a lawn-mowing buisness in her neighborhood. she creats a graph to help her determine what to charge customers per lawn to maximize her profits. she uses c to represent the number of lawns she mows and y to represent her profit in dollars.
The profit is maximum when 40 lawns are mowed.
What is function?A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.
Given is that Kieran is starting a lawn-mowing buisness in her neighborhood. She creates a graph to help her determine what to charge customers per lawn to maximize her profits. She uses {c} to represent the number of lawns she mows and {y} to represent her profit in dollars.
The profit is maximum when 40 lawns are mowed as at this point the the peak of the parabola occurs.
Therefore, the profit is maximum when 40 lawns are mowed.
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Select the expressions that are equivalent to 2(-2m+7)-9m. 14m-13 (-2m+7)2-9m 2(-8m+6m+7)-9m 2(-6m+4m+7)-9m
The expressions that are equivalent to 2(-2m+7)-9m are C: 2(-8m+6m+7)-9m and D: 2(-6m+4m+7)-9m.
To see why, let's simplify the original expression:
2(-2m+7)-9m = -4m+14-9m = -13m+14
Now let's simplify the other expressions:
(-2m+7)2-9m = -4m+14-9m = -13m+14
2(-8m+6m+7)-9m = -16m+12m+14-9m = -13m+14
2(-6m+4m+7)-9m = -12m+8m+14-9m = -13m+14
So we can see that the expressions 2(-8m+6m+7)-9m and 2(-6m+4m+7)-9m are equivalent to the original expression.
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The angle of elevation of the top of the building at a distance of 55 m from its foot on a
horizontal plane is found to be 60°. Find the height of the building rounded to the nearest
tenth of a meter.
The height of the building is _______ meters.
Need help
The height of the building, given the angle of elevation and the distance from the top of the building, is such that he height of the building is 95 meter.
How to find the height ?We shall assume that the building, the distance from the foot and the angle, are in a right - angled triangle.
This means that the height of the building is the opposite side and the given distance is the adjacent side.
The relevant operation would be Tan.
The height of the building would be:
Tan ( 60 ° ) = Height / 55 m
Height = Tan ( 60 ° ) x 55
Height = 95 meters
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what’s the range of this function graph
a. [-2, infinite]
b. (-2, infinite)
c. [2, infinite)
d (- infinite, 2)
answer = c
its not a or b because the range starts at y = 2
W
W
The prism and pyramid above have the same width, length, and height. The volume of the prism is 63 cm³. What is the volume of the
pyramid?
OA. 42 cm³
OB. 189 cm³
OC. 84 cm³
OD.
21 cm³
Vοlume οf the Pyramid is 21 cm³
What is Prism and Pyramid?A prism is a sοlid shape with twο identical parallel bases and flat sides that cοnnect the bases. The sides οf a prism are usually rectangles, but they can alsο be triangles οr οther pοlygοns.
A pyramid is a sοlid shape with a pοlygοnal base and triangular sides that meet at a single pοint called the apex.
When a prism have same base area and height as that οf a pyramid the vοlume οf the prism is three times that οf the pyramid.
=> Vοlume οf prism = 3 Vοlume οf pyramid
Here we have
The prism and pyramid abοve have the same width, length, and height.
The vοlume οf the prism is 63 cm³
As we knοw,
Vοlume οf prism = 3 Vοlume οf pyramid
The vοlume οf the Pyramid = [ Vοlume οf prism ]/ 3
= [ 63 cm³]/3 = 21 cm³
Therefοre,
Vοlume οf the Pyramid is 21 cm³
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Given that tan t = 5. Find tan(t + 71). (Enter an exact number.) Need Help? Read It Submit Answer
The exact number for tan(t + 71) is -0.5846.
To find tan(t + 71), we can use the formula for the sum of two angles:
tan(t + 71) = (tan t + tan 71)/(1 - tan t * tan 71)
Given that tan t = 5, we can substitute this value into the formula:
tan(t + 71) = (5 + tan 71)/(1 - 5 * tan 71)
To find the exact number for tan 71, we can use a calculator or a table of trigonometric values. The exact value of tan 71 is approximately 2.9042.
Substituting this value into the formula, we get:
tan(t + 71) = (5 + 2.9042)/(1 - 5 * 2.9042)
Simplifying the expression, we get:
tan(t + 71) = (7.9042)/(1 - 14.521)
tan(t + 71) = (7.9042)/(-13.521)
tan(t + 71) = -0.5846
Therefore, the exact number for tan(t + 71) is -0.5846.
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Solve the rational equation 5 over X plus X +5 over X +2 equals -11 over X to the 2nd+ 2X
After solving the equation, the solution if the equation is -2 and -11/5.
To solve the rational equation we first express the given statement in the form of mathematical equation. After that we solve the equation.
As the statement is 5 over X + 2. It can be written as 5/(x + 2)
Then the second statement is -11 over X to the 2nd + 2X which is written as -11/(x² + 2x).
There have equal sign between both expression. So the expression is
5/(x + 2) = -11/(x² + 2x)
Now solve it.
Multiply by (x² + 2x) on both side, we get
5(x² + 2x)/(x + 2) = -11
Multiply by (x + 2) on both side, we get
5(x² + 2x) = -11(x + 2)
Now simplify using the distributive property
5x² + 10x = -11x - 22
Add 11x on both side, we get
5x² + 21x = - 22
Add 22 on both side, we get
5x² + 21x + 22 = 0
Now we factor the equation
5x² + (11 + 10)x + 22 = 0
5x² + 11x + 10x + 22 = 0
x(5x + 11) + 2(5x + 11) = 0
(5x + 11)(x + 2) = 0
Now equating equal to 0.
5x + 11 = 0 x + 2 = 0
5x = -11 x = -2
x = -11/5
The solution if the equation is -2 and -11/5.
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The complete question is:
Solve the rational equation 5 over X + 2 equals -11 over X to the 2nd + 2X.
Ver en español
Felix and his friends attended the opening of the new community center at Forest Ridge Park. The mayor unveiled a parallelogram-shaped decorative plaque at the entrance to the park with the date of the special event. Its bottom edge is 9 inches long, and its area is 126 square inches.
Which equation can you use to find how tall the plaque is, h?
How tall is the plaque?
Write your answer as a whole number or decimal. Do not round.
inches
The answer to this question is as follows The plaque measures 14 inches equation tall as a result.
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.
Because we are aware that the plaque has a parallelogram shape, we may determine its height using the formula for a parallelogram's area. The formula for a parallelogram's area is:
Base area x height
In this instance, we are aware that the area is 126 square inches, and the base (the bottom border) is 9 inches. Thus, we can enter those values into the formula to find the height:
126 = 9h
h = 126/9
h = 14
The plaque measures 14 inches tall as a result.
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PLEASEEEE HELP A GIRL OUTTTTTTTT 17 POINTSSSSSSS!!!!!!!!!
Answer: B
Step-by-step explanation:
-2x-4+2x+32-2x-21=18
Answer:
x = -5.5
Step-by-step explanation:
-2x - 4 + 2x + 32 - 2x - 21 = 18
-2x - 4 + 32 - 21 = 18
-2x + 7 = 18
-2x = 11
x = -5.5
So, the answer is x = -5.5
Describe transformation from f(x)=√x +3 to g(x)=f(x)+k
The transformation applied to the function f(x) is a translation of 4 units up.
Which is the transformation applied?Here we can see that:
f(x) = √(x + 3)
and g(x) is a vertical translation of k units:
g(x) = f(x) + k
To get the value of k, just look how many units the graph has been translated up ( remember that each of the smalls squares in the coordinate axis represents a single unit), we can see that it is 4 units up, then k = 4
We have a vertical translation of 4 units up.
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Half Challenge! Taxpayer Name: Bob Tax Bracket Information: Money earned between $0 and $200, 10% tax Money earned between $200 and $500, 20% tax Money earned after $500, 40% tax Finances:
Income: $750 Adjusted Income:
Expenses: $100 Adjusted Expenses: Deductible: $150
Return:
Bob Returned $120 Did Bob commit Tax Fraud? YES NO Bob's Net Income:
Bob did not commit tax fraud based on the given information.
To calculate Bob's net income, we need to start with his adjusted income, which is his income minus his expenses and deductible:
Adjusted Income = Income - Expenses - Deductible
Adjusted Income = $750 - $100 - $150
Adjusted Income = $500
Next, we need to determine the amount of tax that Bob owes based on his tax bracket information:
Tax Owed on first $200 = $200 * 0.10 = $20
Tax Owed on next $300 ($200 to $500) = $300 * 0.20 = $60
Tax Owed on remaining $0 ($500 and above) = $0 * 0.40 = $0
Total Tax Owed = $20 + $60 + $0 = $80
Now, we can calculate Bob's net income by subtracting his tax owed from his adjusted income:
Net Income = Adjusted Income - Tax Owed
Net Income = $500 - $80
Net Income = $420
Finally, we can compare Bob's returned amount to his tax owed to see if he committed tax fraud:
Returned Amount - Tax Owed = $120 - $80 = $40
Since Bob's returned amount is less than his tax owed, he did not commit tax fraud. His net income after taxes is $420.
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can someone help me please
Answer:
ABD = 75 degrees
Step-by-step explanation:
ABD is the same as ABC + CBD
ABC = 30
CBD = 45
30+45 = 75 degrees
Therfore, ABD = 75 degrees
What Is 1 + 1 ?
A. Window
B. Two
C. Eleven
Correct Answer Gets Brainliest!
Answer: B -_-
Step-by-step explanation:
0 0=2
select the 2 linear functions A) (y=6x+14), B) (y= x/4 + 1), C) (y=x^3), D) (y=3/x +2)
The two linear equations are y = 6x + 14 and y = x/4 + 1. Then the correct options are A and B.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Let's check all the options, then we have
A) y = 6x + 14, the equation is a linear equation because the degree is one.
B) y = x/4 + 1, the equation is a linear equation because the degree is one.
C) y = x³, the equation is a cubic equation because the degree is three.
D) y = 3/x +2, the equation is a non-linear equation because the degree is negative one.
The two linear equations are y = 6x + 14 and y = x/4 + 1. Then the correct options are A and B.
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Which of the following are the coordinates of point B’, the image of point B after a translation of (x-4,y-3)?
The coordinates of point B' after the translation are given as follows:
B'(1,-2).
What is a translation?A translation happens when either a figure or a function are moved horizontally or vertically.
The meaning of the translation for this problem is given as follows:
x - 4: 4 units left.y - 3: 3 units down.The coordinates of point B are given as follows:
B(5,1).
Hence the coordinates of point B' are given as follows:
x = 5 - 4 = 1.y = 1 - 3 = -2.Meaning that the last option is correct.
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Given \( f(x)=5 x \) and \( g(x)=7 x^{2}+6 \), find the following expressions. (a) \( (f \circ g)(4) \) (b) \( (g \circ f)(2) \) (c) \( (f \circ f)(1) \) (d) \( (g \circ g)(0) \) (a) (fog)(4) = (Simplify your answer.)
(b) (go f)(2) = (Simplify your answer.)
(c) (fo f)(1) = (Simplify your answer.)
(d) (go g)(0) = (Simplify your answer.)
By following this process, we were able to evaluate each expression and get the answers of 590, 181, 25, and 6 respectively.
a) (fog)(4) = (f(g(4)) = f(7(4)^2+6) = 5(7(16)+6) = 5(118) = 590
b) (gof)(2) = (g(f(2)) = g(5(2)) = 7(5)^2+6 = 7(25)+6 = 181
c) (fof)(1) = (f(f(1)) = f(5(1)) = 5(5) = 25
d) (gog)(0) = (g(g(0)) = g(7(0)^2+6) = 7(0)+6 = 6
To find the expressions given, we need to first use the definition of a function composition. This means that when we have a function \(f\) composed with a function \(g\), this is represented by \(f\circ g\). When we use a function composition, we evaluate the function \(g\) first, then plug the result into the function \(f\).
For example, in part (a) we have the expression \(f\circ g(4)\). We need to first evaluate \(g(4)\), which gives us \(7(4)^2+6\). Then we plug this result into the function \(f\) to get \(f(7(4)^2+6)= 5(7(4)^2+6) = 590\). We can repeat this process for the rest of the parts. For part (b) we have \(g\circ f(2)\). Evaluating \(f(2)\) gives us \(5(2)\), which we then plug into \(g\) to get \(g(5(2))= 7(5)^2+6 = 181\).
For part (c) we have \(f\circ f(1)\), so evaluating \(f(1)\) gives us \(5(1)\) and then plugging this into \(f\) gives us \(f(5(1))= 5(5) = 25\). Finally, for part (d) we have \(g\circ g(0)\), so evaluating \(g(0)\) gives us \(7(0)^2+6\), which we plug into \(g\) to get \(g(7(0)^2+6) = 7(0)+6 = 6\).
In conclusion, we can use function composition to find the expressions given. We start by evaluating the inner function, then plugging the result into the outer function.
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What is the difference in area betwee circle with its of 10 centimeters a square inscribed in it, to the neares whole?
The difference in area between a circle with a radius of 10 centimeters and a square inscribed in it is 114 cm².
The difference in area between a circle with a radius of 10 centimeters and a square inscribed in it can be found by calculating the area of the circle and the area of the square and then subtracting the two.
First, calculate the area of the circle using the formula
A = πr²,
where A is the area and r is the radius.
A = π(10)² = 100π ≈ 314.16 square centimeters
Next, calculate the area of the square. Since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square. The diameter of the circle is 2r, or 20 centimeters.
Using the Pythagorean theorem, we can find the side length of the square:
s² + s² = (20)²
2s² = 400
s² = 200
s ≈ 14.14 centimeters
The area of the square is s² or (14.14)² ≈ 199.97 square centimeters.
Finally, subtract the area of the square from the area of the circle to find the difference:
314.16 - 199.97 ≈ 114.19 square centimeters
To the nearest whole, the difference in area is 114 square centimeters.
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Multiplicative property of equality with whole numbers Solve for u. 78=6u Simplify your answer as much as possible. u
The multiplicative property of equality states that the same number can be added to or multiplied by both sides of an equation to obtain an equivalent equation. In this case, dividing both sides by 6 gives us u = 78/6 = 13.
The multiplicative property of equality states that if two numbers are equal, then multiplying both sides of the equation by the same number will also result in an equation that is still equal. In other words, if a=b, then ac=bc. We can use this property to solve for the variable u in the equation 78=6u.
To isolate the variable on one side of the equation, we can divide both sides by 6. This will give us:
78/6 = 6u/6
Simplifying the equation gives us:
13 = u
So the solution for u is 13.
In conclusion, the multiplicative property of equality with whole numbers was used to solve for the variable u in the equation 78=6u. The solution is u=13.
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Bob's gift shop sold a record number of cards for Mother's Day. One salesman sold 47 cards, which was 4% of the cards sold for Mother's Day. How many cards were sold for Mother's Day?
Answer:
We can use the concept of percentage to solve this problem.
Let's represent the total number of cards sold for Mother's Day by "x". We know that one salesman sold 47 cards, which was 4% of the total number of cards sold. So we can set up an equation:
0.04x = 47
Solving for "x", we divide both sides by 0.04:
x = 47 / 0.04
Using a calculator, we get:
x = 1175
Therefore, the total number of cards sold for Mother's Day was 1175.