Answer: see below
Step-by-step explanation:
To complete the table, we need to evaluate the function f(u) = u² + 2u + 3 for the given values of u.
f(u) U
-20 (-20)² + 2(-20) + 3 = 403
-10 (-10)² + 2(-10) + 3 = 117
0 0² + 2(0) + 3 = 3
10 10² + 2(10) + 3 = 123
The table can be filled as follows:
f(u) U
403 -20
117 -10
3 0
123 10
The function f(u) = u² + 2u + 3 is a quadratic function, which has a parabolic shape. The coefficient of u² is positive, which means the parabola opens upward. The vertex of the parabola is at the point (-1, 2), which is the minimum point of the function.
As u increases or decreases from 0, the function increases. This is reflected in the table, where we see that the values of f(u) increase as u moves away from 0.
The table gives us a set of ordered pairs that represent the function f(u) for the given values of u. These ordered pairs can be plotted on a coordinate plane to draw the graph of the function.
a rancher has 10,000 linear feet of fencing and wants to enclose a rectangular field and then divide it into two equal pastures, with an internal fence parallel to one of the rectangular sides. what is the maximum area of each pasture? round to the nearest square foot.
For a 10,000 linear feet of fencing and wants to enclose a rectangular field, the maximum area of each pasture is equals to the 2,083,333.33 square feet .
A rancher wants to enclose about 10,000 linear feet of fencing in a rectangular field. There are two equal pastures. Let, W the Width and L the Length with 3 pieces. The total length will be equal to 2W + 3L. As we know, the Area of rectangle, A = L×W ---(1)
Perimeter of rectangle = 10,000 feet, so sum of all sides of rectangle, 2W + 3L = 10000
Solve for determining value of L, 3L
= 10000 - 2W
[tex]L = \frac{ 10000 - 2W }{3}[/tex]
Substitutes for L into the first equation, A = L×W
[tex]A = W(\frac{ 10000 - 2W }{3})[/tex]
For maximum area, set the 1st derivative = 0.
differentiating above Area equation w.r.t W,[tex] A = \frac{(10000 \: W - 2W^2)}{3}[/tex]
[tex] \frac{dA}{dW} = \frac{d(\frac{10000 \: W - 2W^2}{3})}{dW}[/tex]
=> [tex]\frac { 1}{3}(10000 - 4W) = 0 [/tex]
=> W = 2500 meters
Now, using above relation, 2W + 3L= 10000
=> 5000 + 3L = 10000
=> 3L = 5000
=> L = 1667 meters
Area = 2500× 5000/3 = 4,166,666.67 sq feet. Now, maximum area of each pasture
= 4,166,666.67/2 = 2,083,333.33333 sq. feet. Hence, required value is 2,083,333.33 sq. feet.
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Let h(x) be the number of hours it takes
a new factory to produce x engines. The
company's accountant determines that
the number of hours it takes depends on
the time it takes to set up the machinery
and the number of engines to be
completed. It takes 6.5 hours to set up
the machinery to make the engines and
about 5.25 hours to completely
manufacture one engine. The
relationship is modeled with the
function b(x)=6.5+5.25%
1. Determine the x and y-
intercepts of the function.
2. Is the function increasing or
decreasing?
Based on the information, the y-intercept is (0, 6.5).
The function is increasing.
How to calculate the interceptBased on the information, we can use the variable x. This will be:
0 = 6.5 + 5.25%x
-6.5 = 5.25%x
x = -6.5 / 5.25% ≈ -123.81
There's no x intercept due to the negative value which doesn't make sense in this scenario.
In order to find the y-intercept, we set x = 0 and evaluate b(x):
b(0) = 6.5 + 5.25% × 0 = 6.5
Therefore, the y-intercept is (0, 6.5).
The function b(x) is increasing since the coefficient of x is positive (5.25%). This means that as the number of engines produced (x) increases, the time it takes to manufacture the engines also increases.
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Write the transformation matrix and the resultant matrix that will translate the triangle, (-2, 4), given the triangles vertices are at A(-1, 3), B(0, -4) and C(3, 3).
The new vertices of the triangle after the translation are A'(-3, 7), B(-2, 0), and C(1, 7), which form the resultant matrix:
|-3 -2 1 |
| 7 0 7 |
| 1 1 1 |
Define translationIn mathematics, a translation refers to a geometric transformation that moves every point in a figure or a space by the same distance in a given direction. In other words, it involves sliding a figure or a point to a new location without changing its size, shape, or orientation.
To translate the triangle by a vector (x, y), we use the following transformation matrix:
|1 0 x|
|0 1 y|
|0 0 1|
For the given triangle with vertices at A(-1, 3), B(0, -4), and C(3, 3), let's assume we want to translate it by a vector (−2, 4). Then the transformation matrix for translation is:
|1 0 -2|
|0 1 4 |
|0 0 1 |
To apply this transformation matrix to each vertex of the triangle, we represent the vertices as column vectors and multiply them by the matrix:
| -1 | | 1 0 -2 | | -3 |
| 3 | -> | 0 1 4 | = | 7 |
| 1 | | 0 0 1 | | 1 |
| 0 | | 1 0 -2 | | -2 |
| -4 | -> | 0 1 4 | = | 0 |
| 1 | | 0 0 1 | | 1 |
| 3 | | 1 0 -2 | | 1 |
| 3 | -> | 0 1 4 | = | 7 |
| 1 | | 0 0 1 | | 1 |
So, the new vertices of the triangle after the translation are A'(-3, 7), B(-2, 0), and C(1, 7), which form the resultant matrix:
|-3 -2 1 |
| 7 0 7 |
| 1 1 1 |
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Drag the tiles to the correct boxes to complete the pairs *Not all tiles will be used*
Match the correct volume formula with each described figure
1. V = 1/2 * pi * x ^ 3;
2. V = x ^ 3;
3. V = 1/3 * x ^ 3;
4. V = 1/2 * pi * x ^ 3;
5. V = 1/2 * x ^ 3;
6. V = pi * x ^ 3
A. a prism with a height of x cm and a square base with side length of x cm
B. a cylinder with a radius of x and height of cm
C. a pyramid with a height of x cm and a square base with a side length of x cm
D. a cone with a radius of x cm and height of x cm
A. a prism with a height of x cm and a square base with side length of x cm = option 2
B. a cylinder with a radius of x and height of cm= option 6
C. a pyramid with a height of x cm and a square base with a side length of x cm = option 3.
How to match the correct formula with the statements given ?For statement A=
The formula for a square based prism= a²h
where a= X
h = X
Vol = x³
For statement B;
The formula for a cylinder=πr²h
where r= X, h= X
Vol= π×x³
For statement C;
The formula for a square based pyramid= 1/3a²h
a = X
H = X
Vol = 1/3 * x³
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What is the correct list of functions ordered from least to greatest by average rate of change over the interval 0 less than or equal to x less than or equal to 3
The correct list of functions ordered from 0 ≤ x ≤ 3 is:
h(x) = sin(x) < f(x) = x^2 < g(x) = 2x + 1 < k(x) = e^x
Line connecting the interval's endpoints in order to compute the average rate of change for each function over range 0 x 3.
For f(x) = x^2, the slope between x = 0 and x = 3 is:
[tex](f(3) - f(0)) / (3 - 0) = (9 - 0) / 3 = 3[/tex]
For g(x) = 2x + 1:
[tex](g(3) - g(0)) / (3 - 0) = (7 - 1) / 3 = 2[/tex]
For h(x) = sin(x):
[tex](h(3) - h(0)) / (3 - 0) = (sin(3) - sin(0)) / 3[/tex] ≈ 0.279
For k(x) = e^x, the slope between x = 0 and x = 3 is:
[tex](k(3) - k(0)) / (3 - 0) = (e^3 - 1) / 3[/tex] ≈ 6.076
Therefore, correct list of functions ordered from least to greatest by average rate of change over the interval 0 ≤ x ≤ 3 is:
[tex]h(x) = sin(x) < f(x) = x^2 < g(x) = 2x + 1 < k(x) = e^x[/tex]
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--The complete Question is, Consider the following four functions:
f(x) = x^2
g(x) = 2x + 1
h(x) = sin(x)
k(x) = e^x
What is the correct list of functions ordered from least to greatest by average rate of change over the interval 0 ≤ x ≤ 3? --
Does the data in this table represent a function?
X -4 2 -4 2 -4
y -3 1 5 8 9
A. No, it is not a function.
B. Yes, a linear function.
C. Yes, a nonlinear function.
Answer:
A
Step-by-step explanation:
It is not a function because both x-values -4 and 2 have multiple y-values. Try the vertical line test if you don't understand.
Explain how you would find the highest point the rider achieved on this jump.
A. What would be the two quantities you would measure for x and y?
B. What would your equation possibly look like? Put the picture on a grid and show some possible points
C. What points could you use to find the maximum height? How would you use those points? Be specific
Answer:
Step-by-step explanation:
36
Find the area.
Area =
6 m
10 m
square meters
Answer:
32m
Step-by-step explanation:
area= L×B
L= 10m ×2 =20m
B= 6m×2 =12m
Area= 20m + 12m = 32m
What is the value of the digit 7 when 2.7 is multiplied by 10 to the 2nd power
Answer:
When 2.7 is multiplied by 10 to the 2nd power, it becomes 270. The digit 7 is in the ones place, so the value of the digit 7 is 7.
Step-by-step explanation:
3. A function can have zero to many parameters, and it can return this many values.a. zero to manyb. noc. only oned. a maximum of tene. None of these
A function can have zero to many parameters, and it can return only one (option c).
In mathematics, a function is a rule that maps each element from one set, called the domain, to a unique element in another set, called the range.
The number of parameters a function can take determines the number of arguments required to call the function.
For instance, a function f(x) takes one parameter, and it requires one argument to be called.
On the other hand, a function g(x,y,z) takes three parameters, and it requires three arguments to be called. However, a function h() takes no parameters and does not require any argument to be called.
Hence the correct option is (c).
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RIGHT ANSWER GETS BRAINLIEST AND 11 POINTS ‼️‼️‼️‼️‼️‼️‼️‼️
Answer:
Less
Step-by-step explanation:
Look at the hundreds number in set L the numbers are greater in Set L than Set K.
HELPPPP ASAPPP WILLL GIVE BRAINLEST
Answer:
2) The smallest amount of fabric Amy can buy is 78 inches = 2 1/6 yards, so she will need 2 1/4 yards of fabric.
3) $2.25(5) + $0.35(2) + $0.25(3) + $0.60 = $13.30 before sales tax
Sales tax is $13.30 × .08 = $1.06
Total is $13.30 + $1.06 = $14.36
Triangulation
Here are two-five pointed stars. Both figures A and B are polygons.They are both composed of line segments and are
two dimensional. Neither have curves. Do you agree with the statement?
A plane is an infinite two-dimensional figure.
We have,
When two planes intersect, they form a line.
The vector equation for the line of intersection is given by
r = r₀ + tₓ
where r₀ is a point on the line
tₓ is the cross product of the normal vectors of the two planes.
The parametric equations for the line of intersection are given by
x = a , y = b and z = c
where a, b and c are the coefficients from the vector equation
r = a (i) + b (j) + c (k)
The line of intersection will be perpendicular to the normal of both the planes
The line of intersection lies on both the planes
Therefore , a line is observed at the intersection of two planes
Given data ,
A line is one-dimensional, a segment is a part of a line and is also one-dimensional, and a point is zero-dimensional.
A plane, on the other hand, is a flat, two-dimensional surface that extends infinitely in all directions.
It has length and width, but no thickness, and can be thought of as an infinitely large sheet of paper.
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complete question:
Which of the following is an infinite two-dimensional figure?
A. A line
B. A plane
C. A segment
D. A point
The central train station in a large city is located at the intersection of tracks A, B, and C as shown below. pls pls pls respond now!!!!
a.) The position of the train moving from station A to station B after 10 minutes 20 km east
b.) The position of the train moving from station A to station B after 10 minutes, if station B is assumed to be the origin is 50 km west
c.) The position of the train moving from station A to station B after 10 minutes, if station C is assumed to be the origin is 80 km west
How do we know?we have that:
Railway station - A B C
Distance(km) - 0 30 60
Starts from A , train reaches B be in 15 minutes
Starts from A , train reaches C be in 30 minutes
a.)
If A is origin
As train covers 30km = 15 minutes
⇒ 15 minutes = 30 km
1 minute = km
⇒ 10 minutes = 2×10 = 20 km
The position of the train moving from station A to station B after 10 minutes = 20 km east
b.)
If B is origin then , the position of the train moving from station A to station B after 10 minutes = 30 + 20 = 50 km west
c.)
If C is the origin then , the position of the train moving from station A to station B after 10 minutes = 60 + 20 = 80 km west
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#possible complete question:
Three railway stations, A, B, and C, are situated on a straight railway route. Station B is at 30 km in the east from station A and station C is at 60 km in east from station A. A train, with a constant average velocity, starts from A and reaches B in 15 minutes, and reaches C in 30 minutes. Assuming that station A is the origin, find the following:
a. The position of the train moving from station A to station B after 10 minutes.
b. The position of the train moving from station A to station B after 10 minutes, if station B is assumed to be the origin.
c. The position of the train moving from station A to station B after 10 minutes, if station C is assumed to be the origin.
4. Find volume. Show work, round to 2 decimal places.
*cone*
Answer: 80
Step-by-step explanation:
5. Graph f(x)=x + 3
x-2
Answer:
1) f(-2) = -2 + 3
2) f(-2) = 1
Austin collected 30 9/10 kg of glass for music going exactly 2/3 of the glass he collected was blue glass. What was the total amount in kilograms of blue class Austin collected
Answer:
20 3/5
Step-by-step explanation:
this is the correct result when 30 9/10 is multipled by 2/3
Solve the simultaneous equation
x² + y² = 1
5x+12y +13=0
The two solutions to the system of equations are (x,y) = (7/5,-3) and (x,y) = (144/65,-12/13).
What is quadratic equation?
it's a second-degree quadratic equation which is an algebraic equation in x.
We can use substitution to solve this system of equations.
First, we solve the second equation for one of the variables, say x:
5x + 12y + 13 = 0
5x = -12y - 13
x = (-12/5)y - (13/5)
Next, we substitute this expression for x into the first equation:
x² + y² = 1
[(-12/5)y - (13/5)]² + y² = 1
Expanding the square and simplifying, we get:
y² + [(144/25)y² + (312/25)y + (169/25)] + y² = 1
(26/5)y² + (312/25)y + (144/25) = 0
Multiplying both sides by 25 to eliminate the fractions:
26y² + 312y + 144 = 0
Dividing by 4 to simplify:
6.5y² + 78y + 36 = 0
Using the quadratic formula:
y = (-b ± sqrt(b² - 4ac)) / 2a
where a = 6.5, b = 78, and c = 36.
Plugging in these values:
y = (-78 ± sqrt(78² - 4(6.5)(36))) / 2(6.5)
y = (-78 ± sqrt(4761)) / 13
y = (-78 ± 69) / 13
y = -3 or -12/13
If y = -3, then substituting into the equation for x gives:
x = (-12/5)(-3) - (13/5) = 7/5
So one solution is (x,y) = (7/5,-3).
If y = -12/13, then substituting into the equation for x gives:
x = (-12/5)(-12/13) - (13/5) = 144/65
So another solution is (x,y) = (144/65,-12/13).
Therefore, the two solutions to the system of equations are (x,y) = (7/5,-3) and (x,y) = (144/65,-12/13).
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What function is represented by the mapping diagram shown?
-2
00
6
2135
A. F(x)= x+3
B. F(x)= 3x
C. F(x) = 2x
D. F(x)=x-3
24600
2
8
Answer:c. F(x)=2x
Step-by-step explanation:
Which is not a legal python statement
a) print("x")
b) 3 + 4 = x
c) x = 3 + 4
d) x = eval(input("Enter x: "))
1/2 + 1/6 least multiple that is the same add using renamed fraction
Answer: 2/3
Step-by-step explanation: 1/2 converted to 6ths is 3/6 add them together and it’s 4/6 simplify that and it’s 2/3
Can you pls help? This is geometry
The equation of the circle is (x - 2)²+ (y - 3)² = 169
Define circleA circle is a two-dimensional form that may be described as a collection of points that are equally spaced apart from a central fixed point. The radius of a circle is the separation between its centre and any other point on the circle. Another way to think of a circle is as the collection of all points at a certain radius from the centre.
The following is the equation for a circle with centres (a, b) and radius r:
(x - a)² + (y - b)² = r²
In this case, the center of the circle is A(2,3), and a point on the circle is B(7,15). The radius of the circle may be calculated using the distance formula:
r = √((7 - 2)² + (15 - 3)²) = √(5² + 12²) = 13
So the equation of the circle is:
(x - 2)² + (y - 3)² = 13²
(x - 2)²+ (y - 3)² = 169
Therefore, the equation of the circle is (x - 2)²+ (y - 3)² = 169
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Hi
6. Sebastian recorded the price of gas each month for 12 months.
a. Draw a trend line on the scatter plot.
b. If the trend continues, what equation can he use to predict
the price of gas in future months?
Gas Price ($)
N
4
O
Gas Price
2 4 6 8 10 12
Month
X
The trend line for the y = 0.5x + 2 is attached accordingly. Note that the the predicted gas price for month 13 would be $8.50.
What is the explanation for the above response?a. To draw a trend line on a scatter plot, you need to perform a linear regression analysis. This involves finding the line of best fit that passes through the data points. The equation for a linear regression line is typically of the form y = mx + b, where y is the dependent variable (gas price in this case), x is the independent variable (month), m is the slope of the line, and b is the y-intercept.
Once you have the equation for the line of best fit, you can plot it on the scatter plot to visualize the trend. The slope of the line will tell you the direction and steepness of the trend (i.e., whether prices are increasing or decreasing, and how quickly).
b. To predict future values using the trend line, you can simply plug in the value of the independent variable (month) for the month you want to predict, and solve for the dependent variable (gas price). For example, if the equation for the trend line is y = 0.5x + 2, and you want to predict the gas price for month 13, you would plug in x = 13 and solve for y:
y = 0.5(13) + 2
y = 6.5 + 2
y = 8.5
So the predicted gas price for month 13 would be $8.50.
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When we add together the NPV and the false omission rate for any test, why is the sum always 100%?
The sum of the NPV and false omission rate for any test will always be 100%.
Explaining why is the sum always 100%?The NPV (Negative Predictive Value) is the proportion of people who test negative for a condition and actually do not have it, while the false omission rate is the proportion of people who have the condition but test negative for it.
When we add together the NPV and the false omission rate for any test, we are essentially considering all the cases where the test result is negative.
The NPV represents the proportion of people who truly do not have the condition and test negative for it, while the false omission rate represents the proportion of people who actually have the condition but test negative for it.
Together, these two values cover all possible cases where the test result is negative, which means that they represent the entirety of the negative results for the test.
Since the sum of all probabilities for an event must always equal 100%, the sum of the NPV and false omission rate must also equal 100%.
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Rewrite 84 + 36 in the form
a(b+c)
where a is the greatest common factor of 84 and 36
The GCF of 84 and 36 is 12.
The GCF of two non-zero integers, x(36) and y(84), is the greatest positive integer m(12) that divides both x(36) and y(84) without any remainder.
GCF of 36 and 84 by Listing Common Factors
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
There are 6 common factors of 36 and 84, that are 1, 2, 3, 4, 6, and 12. Therefore, the greatest common factor of 36 and 84 is 12.
Rewrite in the form of a(b + c), a is the greatest common factor of 84 and 36
84/ 12 = 7
36/12 = 3
So, 12(84 + 36)
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a student performs an experiment where they flip a coin 3 times . if they perform this experiment 200 times, predict the number of repetitions the experiment that will results in exactly two of the three flips landing on tails
We can predict that out of 200 repetitions of the experiment, approximately 75 of them will result in exactly 2 of the 3 flips landing on tails.
Describe Probability?Probability is a branch of mathematics that deals with the study of random events and the likelihood of their occurrence. It is used to quantify the uncertainty associated with a particular event or outcome. In simpler terms, probability is a measure of the likelihood that a certain event will occur, expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.
There are different types of probability, including classical probability, empirical probability, and subjective probability. Classical probability is based on the assumption of equally likely outcomes, while empirical probability is based on actual observations or experiments. Subjective probability, on the other hand, is based on personal judgment or belief.
The probability of getting tails on any single flip of a fair coin is 1/2. Since we are flipping the coin three times, the probability of getting exactly two tails is given by the binomial probability formula:
P(exactly 2 tails) = (number of ways to get 2 tails out of 3) * (probability of getting tails)² * (probability of getting heads)¹
The number of ways to get 2 tails out of 3 is given by the binomial coefficient "3 choose 2", which is 3. So we have:
P(exactly 2 tails) = 3 * (1/2)² * (1/2)¹ = 3/8
This means that the probability of getting exactly 2 tails in one trial of flipping the coin 3 times is 3/8.
To predict the number of repetitions out of 200 that will result in exactly 2 tails, we can multiply the probability of getting exactly 2 tails in one trial by the total number of trials:
Number of repetitions with exactly 2 tails = P(exactly 2 tails) * total number of trials
Number of repetitions with exactly 2 tails = (3/8) * 200
Number of repetitions with exactly 2 tails = 75
Therefore, we can predict that out of 200 repetitions of the experiment, approximately 75 of them will result in exactly 2 of the 3 flips landing on tails.
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suppose that ann selects a ball by first picking one of two boxes at random and then selecting a ball from this box. the first box contains three orange balls and four black balls, and the second box contains seven orange balls and four black balls. what is the probability that ann picked a ball from the second box if she has selected an orange ball? (enter the value of the probability in decimal format and round the final answer to two decimal places.)
The probability that Ann picked a ball from the second box given that she selected an orange ball is 0.56 or 56% (rounded to two decimal places).
We can use Bayes' theorem to calculate the probability that Ann picked a ball from the second box given that she selected an orange ball.
Let A be the event that Ann selected an orange ball, and B be the event that Ann picked a ball from the second box. We want to find P(B|A), the probability that Ann picked a ball from the second box given that she selected an orange ball.
We know that there are two boxes, each with a probability of 1/2 of being selected. The probability of selecting an orange ball from the first box is 3/7, and the probability of selecting an orange ball from the second box is 7/11. Therefore, the probability of selecting an orange ball overall is:
P(A) = P(A|B)P(B) + P(A|B')P(B')
= (7/11)(1/2) + (3/7)(1/2)
= 25/42
Now we can use Bayes' theorem:
P(B|A) = P(A|B)P(B)/P(A)
= (7/11)(1/2)/(25/42)
= 14/25
Therefore, the probability that Ann picked a ball from the second box given that she selected an orange ball is 0.56 or 56% (rounded to two decimal places).
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Find the area of the figure. Round to the nearest tenth, if necessary.
From the given dimensions, area of the figure is approximately equal to 75.8 cm².
What is a regular polygon?A regular polygon is a closed shape with straight sides and equal-length edges, as well as equal angles between those sides. Examples of regular polygons include equilateral triangles, squares, and hexagons. The number of sides a regular polygon has is referred to as its order, while the measure of each interior angle of a regular polygon can be calculated using the formula (n-2) x 180/n, where n represents the number of sides.
From the image we can see that, ABHI is a square with sides 6 cm, BCDGH is a regular pentagon with sides 6 cm (BH = AI = 6 cm) and DEFG is a rectangle with length 11 cm and breadth 6 cm ( DG = AI = 6 cm).
To find the area of the figure we have to find the area of each shape and add it together.
Area of square = side × side = 6 × 6 = 36 cm²
Area of regular pentagon = [tex]\frac{1}{4} \sqrt{5(5+25)a^{2} }[/tex]
= [tex]\frac{1}{4} \sqrt{5(5+25)6^{2} }[/tex]
= [tex]\frac{1}{4} \sqrt{25 + 10*1.41*36}[/tex]
= [tex]\frac{1}{4} \sqrt{532.6 }[/tex]
≈ 5.77 cm²
Area of rectangle = 2(l + b) = 2(11 + 6) = 34 cm²
Therefore area of the figure = 36 cm² + 5.77 cm² + 34 cm² ≈ 75.8 cm².
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help me asap
pls i need to find z this R.4 ixl for special right triangles
Answer:
3 cm
Step-by-step explanation:
because the correct answer is 3cm
Change the function f(x)=2x^2+4x+3 into vertex form.
Answer:
f(x) = 2(x + 1)² + 1
Step-by-step explanation:
the equation of a parabola in vertex form is
f(x) = a(x - h)² + k
where (h, k ) are the coordinates of the vertex and a is a multiplier
given
f(x) = 2x² + 4x + 3 ← factor out 2 from the first two terms
= 2(x² + 2x) + 3
using the method of completing the square
add/subtract ( half the coefficient of the x- term )² to x² + 2x
f(x) = 2(x² + 2(1)x + 1 - 1) + 3
= 2(x + 1)² - 2(1) + 3
= 2(x + 1)² - 2 + 3
= 2(x + 1)² + 1 ← in vertex form