The mean temperature increases, the electricity bill decreases.
What is the scatter plots?
A scatter plot is a type of graph that is used to display the relationship between two continuous variables.
To find the correlation coefficient, we first need to plot the data points on a scatter plot. We can use a graphing calculator to do this.
Here are the steps:
Press the STAT button on your calculator.Press the EDIT button to enter the data into lists.Enter the Mean Temperature values into L1 and the Electricity Bill values into L2.Press the STAT button again, then select CALC.Choose option 4: LinReg (ax+b).Enter L1 as the X List and L2 as the Y List.Make sure the Freq setting is set to 1.Press ENTER to calculate.The calculator will display the equation of the line of best fit, as well as the correlation coefficient (r).
Using these steps, we get a correlation coefficient of approximately -0.83. Rounded to the nearest hundredth, the correlation coefficient is -0.83.
This negative correlation coefficient indicates a strong negative relationship between mean temperature and electricity bill.
Hence, the mean temperature increases, the electricity bill decreases.
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The number of hours of daylight on a given day in City A is given by the following function, where x is the number of days after January 1.
a. The amplitude of the function y = 4 sin [(2π / 365)(x - 79)] + 12 is 4.
b. The period of the function y = 4 sin [(2π / 365)(x - 79)] + 12 is 365 days.
c. There are approximately 16 hours of daylight on the longest day of the year.
d. There are approximately 8 hours of daylight on the shortest day of the year.
e. The graph for the function is obtained.
What is a function?
In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
The function is given as y = 4 sin [(2π / 365)(x - 79)] + 12.
a. The amplitude of the function is the coefficient of the sine function, which is 4.
Therefore, the amplitude of the function is 4.
b. The period of the function is given by the formula: period = (2π/b), where b is the coefficient of x in the sine function.
In this case, b = (2π/365), so the period is -
period = (2π / (2π/365)) = 365
Therefore, the period value is 365 days.
c. The longest day of the year occurs on the summer solstice, which is around June 21.
On this day, x = 171 (assuming January 1 is x = 1). Plugging x = 171 into the function gives -
y = 4 sin [(2π/365)(171 - 79)] + 12 = 15.998 hours
Therefore, the number of hours for daylight is 16 hours.
d. The shortest day of the year occurs on the winter solstice, which is around December 21.
On this day, x = 355. Plugging x = 355 into the function gives -
y = 4 sin [(2π/365)(355 - 79)] + 12 = 8.002 hours
Therefore, the number of hours for daylight is 8 hours.
e. To graph the function for one period, we need to find the values of y for x ranging from 1 to 365.
We can use a graphing tool or software to plot the function.
The graph is plotted.
The x-axis represents the number of days after January 1, and the y-axis represents the number of hours of daylight.
The function oscillates between 8 and 16 hours of daylight over the course of one year, with the maximum occurring around day 171 and the minimum occurring around day 355.
The graph shows that the function is periodic, with a period of 365 days, as expected.
Therefore, the function is plotted.
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HELP ASAP, WILL GIVE FIRST ANSWER BRAINLIEST, 50 POINTS!!!
Triangle NMO is drawn with vertices N(−4, −2), M(−1, −1), O(−4 , −5). Determine the image coordinates of N′M′O′ if the preimage is translated 7 units to the left.
N′(3, −2), M′(6, −1), O′(3, −5)
N′(−4, −9), M′(−1, −8), O′(−4, −12)
N′(−4, 5), M′(−1, 6), O′(−4, 2)
N′(−11, −2), M′(−8, −1), O′ (−11, −5)
Answer: Your welcome!
Step-by-step explanation:
N′(−11, −2), M′(−8, −1), O′ (−11, −5)
The image coordinates of N′M′O′ if the preimage is translated 7 units to the left are N′(−11, −2), M′(−8, −1), O′ (−11, −5). This is calculated by subtracting 7 from the x-coordinate of each point in the original triangle NMO. For example, the x-coordinate of point N in the original triangle is -4, so the x-coordinate of point N′ in the translated triangle is -4 - 7 = -11. The y-coordinate of point N does not change, so the y-coordinate of point N′ is also -2.
Answer:
To translate the triangle 7 units to the left, we need to subtract 7 from the x-coordinates of each vertex:
N' = (N_x - 7, N_y) = (-4 - 7, -2) = (-11, -2)
M' = (M_x - 7, M_y) = (-1 - 7, -1) = (-8, -1)
O' = (O_x - 7, O_y) = (-4 - 7, -5) = (-11, -5)
Therefore, the image coordinates of N'M'O' are (-11, -2), (-8, -1), and (-11, -5).
So the answer is N′(−11, −2), M′(−8, −1), O′ (−11, −5).
A petri dish of bacteria grow continuously at a rate of 200% each day. If the petri dish began with 10 bacteria, how many bacteria are there after 5 days? Use the exponential growth function f(t) = aert, and give your answer to the nearest whole number.
Answer: 320
Step-by-step explanation:
Multiply each time by 2 so 10*2=20 then 40, 80, 160, and 320
Identify the factor pair of ac you could use to rewrite b to factor the trinomial by grouping.
3x²2² +11x-4
The factors of ac are
(Use a comma to separate answers as needed.)
(x+4), (3x-1) are the required factors in the given trinomial solution.
The solution provided below has been developed in a clear step by step manner.
Step: 1
Given polynomial is 3x²2² +11x-4
=3x(x+4)-1(x+4)
=(x+4)(3x-1)
Explanation: Please refer to solution in this step.
Step: 2
(x+4), (3x-1) are the required factors
A factor is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product because the product is divisible by them. Factors can be found using either division or multiplication.
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The median for the given set of six ordered data values is 31.5.
9 12 23 41 50
What is the missing value?
-
The missing value is
Answer:
Step-by-step explanation:
25
P (5,-4) and Q (-1,-2) are points on a straight line. Find the equation of the perpendicular
bisector of PQ; giving the answer in the form y=mx+c.
To find the equation of the perpendicular bisector of PQ, we need to first find the midpoint of PQ, and then determine the slope of the line perpendicular to PQ that passes through this midpoint.
The midpoint of PQ is given by the formula:
((x1 + x2) / 2, (y1 + y2) / 2)
where (x1, y1) = (5, -4) and (x2, y2) = (-1, -2)
So, the midpoint of PQ is:
((5 - 1) / 2, (-4 - 2) / 2) = (2, -3)
Now, the slope of the line PQ can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = (5, -4) and (x2, y2) = (-1, -2)
So, the slope of PQ is:
(-2 - (-4)) / (-1 - 5) = 2/3
Since the perpendicular bisector of PQ will have a slope that is the negative reciprocal of the slope of PQ, the slope of the perpendicular bisector is:
-1 / (2/3) = -3/2
Finally, we can use the point-slope form of the equation of a line to find the equation of the perpendicular bisector. We know that the line passes through the point (2, -3), and has a slope of -3/2. So, the equation of the line is:
y - (-3) = (-3/2)(x - 2)
Simplifying this equation, we get:
y + 3 = (-3/2)x + 3
y = (-3/2)x + 0
Hence, the equation of the perpendicular bisector of PQ is y = (-3/2)x.
Wyatt earns a base salary of $250 per week and 3% commission on sales. If he sells $10,000 worth of goods this week, what would his paycheck be
Wyatt's paycheck for the week would be $550.
What is Algebraic expression ?
Algebraic expression can be defined as combination of variables and constants.
To calculate Wyatt's paycheck, we need to determine his total earnings for the week.
His earnings from his base salary are simply $250.
To calculate his earnings from commission, we need to determine 3% of his sales.
3% of $10,000 is:
0.03 * 10000 = $300
So Wyatt's earnings from commission would be $300.
Adding his base salary and commission together, we get:
$250 + $300 = $550
Therefore, Wyatt's paycheck for the week would be $550.
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Please help!!
Complete the identity
cos^2 θ/2=?
One identity that we can write here is:
cos²(θ/2) = - sin²(θ/2) + 1
How to complete the identity?Here we want to find something that is equal to cos²(θ/2) for every value of theta, that will be an identity.
The general identity for trigonometric functions is:
cos²(θ) + sin²(θ) = 1
So we can rewrite this as:
cos²(θ) = - sin²(θ) + 1
If instead of theta, the argument is theta over two, we could rewrite this as:
cos²(θ/2) = - sin²(θ/2) + 1
That is the identity.
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Orlando went to a game room that charged $3 admission, plus 0.50 per token. The equation which represents his total cost is y=0.50x + 3. What are the ordered pairs for the equation when you use these x-values: 5, 10, 20?
In the equation, the ordered pair for x = 5 is (5, 5.5), the ordered pair for x = 10 is (10, 8), the ordered pair for x = 20 is (20, 13).
What does a linear equation look like in the slope-intercept form?A linear equation has the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. The line's rate of change, or how steeply it rises or descends as we go down the x-axis, is indicated by the slope. The line's intersection with the y-axis, or the value of y when x is equal to 0, is represented by the y-intercept.
The given equation is:
y=0.50x + 3
Substituting the different values of x we find the required ordered pair as follows:
When x = 5:
y = 0.50(5) + 3 = 5/2 + 6/2 = 11/2 = 5.5
When x = 10:
y = 0.50(10) + 3 = 5 + 3 = 8
When x = 20:
y = 0.50(20) + 3 = 10 + 3 = 13
Therefore, the ordered pair for x = 5 is (5, 5.5), the ordered pair for x = 10 is (10, 8), the ordered pair for x = 20 is (20, 13).
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Suppose you know that the radius of one circle is three times the radius of another circle. Explain how you can determine the ratio of the areas of the circles and the ratio of the circumferences of the circles without knowing the actual measurements.
The ratio for the area and the ratio for the circumferences of the circles are 9 : 1 and 6 : 1 respectively.
What is a circle?
A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle has rotational symmetry around the centre.
If the radius of one circle is three times the radius of another circle, then we can use this information to determine the ratio of the areas of the circles and the ratio of the circumferences of the circles without knowing their actual measurements.
Let's call the radius of the smaller circle "r" and the radius of the larger circle "3r".
Ratio of areas -
The area of a circle is proportional to the square of its radius, so the ratio of the areas of the two circles is -
= (area of larger circle) / (area of smaller circle)
= (π(3r)²) / (πr²)
= 9r²/r²
= 9/1
Therefore, the ratio of the areas of the two circles is 9:1.
Ratio of circumferences -
The circumference of a circle is proportional to its radius, so the ratio of the circumferences of the two circles is -
= (circumference of larger circle) / (circumference of smaller circle)
= 2π(3r) / 2πr
= 6r/r
= 6/1
Therefore, the ratio of the circumferences of the two circles is 6:1.
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For a certain are given as to. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
Since odds in favor of a certain horse finishing in second place are given as 49 to 51 so number of outcomes in favor is m = 49
and number of outcomes not in favor n = 51
and total outcomes are m+n = 49 +51 = 100
The probability of a certain horse finishing in second place is 49 /100 = 0.49.
What is probability?The number of favourable outcomes to all possible outcomes of an event is the ratio, which is known as probability. For an experiment with 'n' number of outcomes, the symbol x can be used to indicate the number of excellent outcomes. The probability of an event can be calculated using the following equation:
x/n, where Probability is Favorable Outcomes/Total Results (Event)
The number of outcomes in favour is m = 49, the number of outcomes not in favour is n = 51, and the total outcomes are m+n = 49 + 51 = 100 because the chances in favour of a particular horse placing in second place are provided as 49 to 51.
The likelihood that a certain horse will place second is 49/100, or 0.49.
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The complete question is:
For a certain are given as to. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
In a board game, Maliq gets x points for landing on a red space, 2x points for landing on a blue
space, and 5x points for going all the way around the board. He lands on 21 red spaces, 19 blue
spaces, and goes around the board twice, for a total of 21x + 19 • 2x + 2 • 5x points. Which
expression also represents Maliq's total points?
a. (21+19) 2x + 10x
b. 42x
C. 69x
d. (21+19+2)(x + 2x + 5x)
The expression that represents Maliq's total points is given as follows:
C. 69x.
How to obtain the expressions?The amount of points for each landing is given as follows:
Red space: x points.Blue space: 2x points.All the way: 5x points.The number of times that Maliq landed on each space is given as follows:
21 red spaces.19 blue spaces.2 around the board.Hence the expression for the total points is given as follows:
21x + 19(2x) + 2(5x) = 21x + 38x + 10x = 69x.
(we multiply the terms and then combine the like terms, as all the terms have the variable x they are like terms).
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What is the diffrence between b and 9
Answer: b-8
Step-by-step explanation:
b-9=b-9
b-9+1=b-8
Determine the exact surface area of the cylinder in terms of TT.
39 1/4
14 1/16
12 1/2
10 15/16
The surface area of the cylinder is 39 1/4
What is a Cylinder?Cylinder is is a three-dimensional solid consisting of two parallel circular bases joined together by a curved surface at a particular distance from the center of the circular bases. Cylinder has two flat ends in the shape of circles.
How to determine the surface area of cylinder
Surface area of cylinder is is the total space occupied by the flat circular bases of the cylinder and the curved surface.
Surface area = 2πrh + 2πr^2
Surface area = 2πr(r + h)
Where π = 22/7
r = 1 1/4 = 5/4
h = 3 3/4 = 15/4
SA = 2 * 22/7 * 5/4(5/4 + 15/4)
SA = 44/7 * 5/4(5)
SA = 44/7 * 6.25
SA = 39.2857 cm^2
Surface Area = 39 1/4 cm^2 is also related to 39.2857 cm^2
Therefore, the exact surface area of cylinder is 39 1/4 cm^2
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Refer to the sample data for pre-employment drug screening shown below. If one of the subjects is randomly selected, what is the probability that the test result is a false positive? Who would suffer from a false positive result? Why?
Pre-Employment Drug Screening Results
Positive test result
Negative test result
Drug Use Is Indicated
Drug Use Is Not Indicated
Subject Uses Drugs
46
8
Subject Is Not a Drug User
1
30
Question content area bottom
Part 1
The probability of a false positive test result is enter your response here.
(Round to three decimal places as needed.)
The probability of a false positive test result is approximately 0.267 or 0.267 probability units.
What is probability?Probability is the measure of the likelihood or chance that an event will occur, expressed as a number between 0 and 1.
For example, the probability of rolling a six on a fair dice is 1/6 or approximately 0.167. It is a fundamental concept in statistics and plays a crucial role in many real-life applications, such as decision-making, risk analysis, and scientific research.
To find the probability of a false positive test result, we need to calculate the proportion of individuals who test positive but do not use drugs out of the total number of individuals. This is represented by the ratio of the number of false positives to the total number of negative test results, which is:
(false positives) / (negative test results) = 8 / 30 = 0.267
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PLEASE HELP ASAP !!!!!!
OPTIONS : YES OR NO
Answer:
Below
Step-by-step explanation:
Yes ..... multiply 'x' by '2' then add 3 to get the 'y' value
this is a line of the equation y = 2x + 3 LINEAR
Express the following expression as a function of an acute angle less than 45°:
cot (-859°)
cot (-859°) can be expressed as cot 41° an acute angle which is less than 45°
What is trigonometry?Trigonometry is used in navigation directions. Estimate the direction you need to place your compass to get a straight line. With the help of a compass and navigation trigonometric functions, it's easy to find a place or find the distance to see the horizon.
What is mathematical expression?A Mathematical Statement which contains numbers and variables.
Angle: The two straight lines meeting at the vertex(a common point)
According to the question:
cot (-859) can be expressed as:
cot (-859 + 360x2)= cot (-859 +720)= cot (139)= cot (180 -139)= cot 41°(or)cot(-859) = cot(-859 + 180°x5)= cot (859 +900)= cot (41)°
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True or False. The point (-1/5, -3/2)
Lies on the graph of f(x).
True; the point (-1/5, -3/2) does lie on the graph of f(x) = 7.5x.
How to determine the true statementFrom the question, we have the following parameters that can be used in our computation:
f(x) = 7.5x
The point is given as
(x, y) = (-1/5, -3/2)
To determine whether the point (-1/5, -3/2) lies on the graph of f(x) = 7.5x, we can substitute x = -1/5 into the equation and see if it gives us y = -3/2.
f(x) = 7.5x
By substitution, we have
f(-1/5) = 7.5(-1/5)
Evaluate
f(-1/5) = -1.5
Express as fraction
f(-1/5) = -3/2
So, the point (-1/5, -3/2) does lie on the graph of f(x) = 7.5x.
Hence, the statement is false.
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Complete question
True or False. The point (-1/5, -3/2) lies on the graph of f(x).
Where f(x) = 7.5x
How can I show my work for 7.540 - 4.826
some students have a goal of collecting 200 leaves for a science project. what does the 0% mean in this situation?what does 100% mean
0% would indicate that the kids have not gathered any leaves for the scientific project, and 100% would indicate that they have succeeded in collecting 200 leaves.
Why is percentage important?In mathematics, a percentage is a quantity or a ratio that is expressed as a fraction of 100. Calculating a percentage involves multiplying a quantity by 100 and dividing it by the total.
To determine "how much" or "how many," utilise the percentage. Calculating the precise quantity or figure that is being discussed is made easier with the use of a percentage. We compare fractions. determining a percentage rise or fall aids in calculating profit and loss margins.
In this case, 0% would indicate that the kids have not gathered any leaves for the scientific project, and 100% would indicate that they have succeeded in collecting 200 leaves for the project.
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find the x intercept and the y intercept for 6x-3y=-12
The x intercept and the y intercept for 6x - 3y = -12 is x = 0.5y + 2 and y = 2x - 4.
What are intercepts in graphs?The term “intercept” refers to the location where a line or curve crosses a graph's axis. The x-intercept is the point at which the x-axis is crossed. The y-intercept is the point at which the y-axis is crossed.
The line's steepness is measured by its slope (m). It is the difference between the y-coordinate change and the corresponding x-coordinate change. For 6x - 3y = 12, just enter the above coordinates and solve for m to obtain the slope.
To solve for x, we solve the equation so the variable x is by itself on the left side:
6x - 3y = 12
x = 0.5y + 2
To solve for y, we solve the equation so the variable y is by itself on the left side:
6x - 3y = 12
y = 2x - 4
Therefore, the x = 0.5y + 2 and y = 2x - 4.
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Gwen purchased a fruit salad for $12 and two
watermelons to cut up and add to the fruit salad.
The watermelons cost the same amount, and all
the fruit was less than $22 in all. Write an
inequality to represent this situation. Then
complete the sentence.
Answer:27
6574
Step-by-step explanation:
Answer:
2x+12<22
Step-by-step explanation:
Bc I got it correct
Select the correct answer from each drop-down menu. The general form of the equation of a circle is 7x2 + 7y2 − 28x + 42y − 35 = 0. The equation of this circle in standard form is . The center of the circle is at the point , and its radius is units.
Answer:
The center of the circle is at the point (2, -3), and its radius is 4 units
Step-by-step explanation:
The equation of the circle in standard form can be obtained by completing the square for both x and y terms as follows:
7x^2 - 28x + 7y^2 + 42y - 35 = 0
7(x^2 - 4x) + 7(y^2 + 6y) = 35
7(x^2 - 4x + 4 - 4) + 7(y^2 + 6y + 9 - 9) = 35
7[(x - 2)^2 - 4] + 7[(y + 3)^2 - 9] = 35
7(x - 2)^2 + 7(y + 3)^2 - 112 = 0
7(x - 2)^2 + 7(y + 3)^2 = 112
Dividing by 112 on both sides, we get:
(x - 2)^2 + (y + 3)^2 / 16 = 1
Therefore, the equation of the circle in standard form is (x - 2)^2 + (y + 3)^2 / 16 = 1.
The center of the circle is at the point (2, -3), and its radius is 4 units.
Find the equation of the line shown.
I need it quick pls
The equation of the line shown is y = 2x.
What is a proportional relationship?In Mathematics, a proportional relationship can be defined as a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:
y = kx
Where:
x and y represents the variables or data points.k represents the constant of proportionality.Next, we would determine the constant of proportionality (k) as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 2/1 = 4/2
Constant of proportionality, k = 2.
Therefore, the required equation is given by:
y = kx
y = 2x
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1. The circumcenter of ARST is A. What is the length of AT?
The correct option is D. For the circumcenter A of the triangle ∆RST, the length of AT is equal to 23.
What is the circumcenter of a triangleThe circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. This point is equidistant from the three vertices of the triangle and is the center of the circumcircle, which is the circle passing through all three vertices of the triangle.
Hence;
AS = AR = AT
3x - 7 = 2x + 3
3x - 2x = 7 + 3 {collect like terms}
x = 10
so;
AS = 3(10) - 7
AS = 30 - 7
AS = 23 and AT also is = 23.
Thus , for the circumcenter A of the triangle ∆RST, the length of AT is equal to 23.
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PLEASE HELP
Using the recursive formula given, find the first five terms of each sequence.
1.) A1 = 12, An=3An-1 -21, n≥2
2.) A1 = 1/2, An = An-1 + 3/2, n≥2
The first five terms of the given sequence are, 12, -9, 30, -51, -72....
What is a sequence?A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.
Given is the recursive formula, we need to find the first five terms of the sequence,
1) a₁ = 12, aₙ = 3 and aₙ₋₁-21 :-
a₂ = a₂₋₁ -21
= 12-21
= -9
a₃ = a₃₋₁-21
= -9-21
= -30
a₄ = a₄₋₁-21
= -30-21
= -51
a₅ = a₅₋₁-21
= -51-21
= -72
Hence, the first five terms of the given sequence are, 12, -9, 30, -51, -72....
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2
The model represents the quotient of two decimals.
PLEASE LOOK AT PIC TO DETERMINE
Which expression does this model represent?
A. 0.875 divided by 3.5
B. 3.5 divided by 0.875
C. 4.0 divided by 0.875
D. 0.875 divided by 4.0
Answer:
The model represents the division of the larger rectangle (3.5 units wide) into 4 equal parts, and shading 1 part of it.
This means that the division is 3.5 divided into 4 parts, with each part being equal to the smaller rectangle with a width of 0.875.
Therefore, the expression represented by the model is:
B. 3.5 divided by 0.875
Step-by-step explanation:
The scale of the model is 1 inch = 9 feet. The fountain at the center of the city is 18 1/2 feet wide. How wide is the fountain in the model?
Answer: 2.05 repeating
Step-by-step explanation:
18/2= 2
1/2 feet goes into 9 18 times so 1/18 = .055 repeating
The figure shows a large square with an area of a^2 that contains a smaller square with an area of b^2. Describe the regions that represent a^2-b^2.
As a result, the total area of these four triangular regions, a2 - b2, represents the disparity between the two squares' surface areas.
what is triangle ?A triangle is a two-dimensional geometric shape with three straight edges and three angles. It is one of the fundamental geometric forms and can be categorised in various ways based on the sides and angles it has. Triangles are typically categorised according to their edges as follows: Equilateral triangles have three identical sides and three equal angles, each measuring 60 degrees. A triangle with an isosceles triangle has two edges that are equal and two angles that are equal.
A triangle called a scalene triangle lacks equal edges and angles.
given
The four triangular areas outside the smaller square but inside the bigger square are the regions that stand for a2 and b2.
Consider the size of the large rectangle, which is a2, to understand why this is the case. Consider the smaller square's size, which is b2.
These two areas vary by (a2) - (b2), which we can also write as:
(a + b)(a - b) (a - b)
According to this formula, the difference between the two squares' areas is equivalent to the product of their side lengths' sum and differences.
We can write the sum and difference of their side lengths as (a + b) and (a - b), respectively, because the big square's side length is a and the smaller square's side length is b.
The area of the four triangular areas outside the smaller square but inside the larger square is represented by the product (a + b)(a - b).
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The fencing of the left border costs $8 per foot, while the fencing of the lower border costs $2 per foot. (No fencing is required along the river.) You want to spend $320 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose?
The dimensions of the garden are 20 feet by 80 feet, and the area enclosed is 1600 square feet.
How is optimization employed in mathematics?
The process of selecting the optimal answer from a range of alternatives is known as optimization. In mathematics, optimization is employed to determine a function's maximum or least value. Typically, an optimization issue entails the optimization of a function under a set of restrictions.
Let's assume that the length of the left border is x feet.
The length of the lower border is y feet.
From the given situation the equation is:
8x + 2y = 320
To maximize the area of the garden, we maximize the product xy.
2y = 320 - 8x
y = 160 - 4x
Substituting this expression for y into the expression for the area, we get:
A = xy = x(160 - 4x) = 160x - 4x²
To find the value of x that maximizes the area, we can take the derivative of A with respect to x and set it equal to zero:
dA/dx = 160 - 8x = 0
x = 20
Substituting x = 20 back into the expression for y, we get:
y = 160 - 4x = 80
Therefore, the dimensions of the garden are 20 feet by 80 feet, and the area enclosed is: A = xy = (20)(80) = 1600 feet².
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