We require a more accurate and objective measure to determine the association between the two variables because visual exams are primarily subjective. We employ the linear correlation coefficient to express the degree and direction of the association between two variables:
[tex]r=\frac{\sum \frac{\left(x_i-\bar{x}\right)}{s_x} \frac{\left(y_i-\bar{y}\right)}{s_y}}{n-1}[/tex]
where the sample means and standard deviations of the x's are x and sx, respectively, and the y's are y and sy, respectively. n is the sample size. .
A different method of calculating the correlation coefficient is:
[tex]r=\frac{S_{x y}}{\sqrt{S_{x x} S_{y y}}}[/tex]
[tex]\begin{aligned}& \text { where } S_{x x}=\sum x^2-\frac{\left(\sum x\right)^2}{n} \\& S_{x y}=\sum x y-\frac{\left(\sum x\right)\left(\sum y\right)}{n} \\& S_{y y}=\sum y^2-\frac{\left(\sum y\right)^2}{n}\end{aligned}[/tex]
In honor of Karl Pearson, who created the linear correlation coefficient in the first place, it is also known as Pearson's product moment correlation coefficient. This metric quantifies the strength of the linear, or straight-line, relationship between the two variables as well as its positive or negative direction.
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anyone know how to do this
From the triangle, the values of a and b are 5.1 units and 4.4 units respectively
How to find the lengths of a triangle?You should know that a triangle is a polygon with three sides, three vertices, three angles whose sum of angles is 180⁰
Using the trig ratio of Cosine
CosY=Adjacent/Hypothenuse
cos31=w/6
Cross multiplying we have
6*cos31=w
w=5.1 units
Then to find b, note that /VW/=/VY/=5.1
<VYX=<VWX............... base angles of isosceles triangle WVY
Using cosine again from trig.
Cos31=b/5.1
5.1*cos31=b
simplify to get b=4.4 units
Therefore b= 4.4 units
a=90-31 (remaining 2 angles of a right angled triangle)
a=59⁰
In conclusion, the value of a=59⁰ and the value of b=4.4 units
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I need the answer now
Please help me
I’ll give you the Brainiest
There is a close relationship between the air pressure inside a hurricane and its maximum sustained wind speed: Y-1.15x + 1190 where
x is the air pressure in millibars (kPa) and y is the wind speed in knots (nautical miles per hour). Using the model, what would be the wind
speed of a hurricane with an air pressure of 956 kPa? Round your answer to the nearest knot
A) 50 knots
B) 75 knots
C) 115 knots
D) 91 knots
The wind speed of a hurricane with an air pressure of 956 kPa will be 2289.4 knots.
What is a mathematical function, equation and expression?function : In mathematics, 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.expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators.equation : A mathematical equation is used to equate two expressions.Given is that relationship between the air pressure inside a hurricane and its maximum sustained wind speed → y = 1.15x + 1190, where [x] is the air pressure in millibars (kPa) and [y] is the wind speed in knots (nautical miles per hour).
The air pressure inside a hurricane and its maximum sustained wind speed are related as → y = 1.15x + 1190
For x = 956 kpa of pressure, we can write -
y = 1.15 x 956 + 1190
y = 2289.4 knots
Therefore, the wind speed of a hurricane with an air pressure of 956 kPa will be 2289.4 knots.
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look at the pic and answer
2/3 + 1/5
4/5 + 1/2
5/7 + 1/3
1/4 -2/5
Answer:
A. 13/15
B. 13/10
C. 22/21
D. −3/20
Step-by-step explanation:
Just make the denominators equal to each other and solve. Ask your teacher for extra help
g what would need to be known about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solutionwhat would need to be known about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution
There are no free variables for the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution.
Define the term unique solution?Identical to a coefficient matrix, an augmented matrix has an additional column that contains the values from the right side of a linear system of equations. A number of equations must match the number of unknowns if a system of equations does have a single unique solution (variables).For the stated question-
It is necessary to know that a system is consistent (i.e., there is no pivot in the last column of the augmented matrix) and that there are no free variables in order to determine whether it has a unique solution (like a pivot position in every column of the coefficient matrix).
Also take note that this rules out having fewer rows than columns.
Thus, there are no free variables for the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution.
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A 8-inch candle burns down in 4 hours. Assuming the candles are the same thickness and make (that is, directly proportional), how long would it take a 12-inch candle to burn down?
Answer:
To determine how long it would take a 12-inch candle to burn down, we need to first determine the relationship between the length of the candle and the time it takes to burn down. Since the candles are the same thickness and make, we can say that the time it takes for a candle to burn down is directly proportional to its length. This means that we can write an equation to represent the relationship between the length of the candle and the time it takes to burn down as follows:
t = k * l
where t is the time it takes for the candle to burn down, k is the proportionality constant, and l is the length of the candle.
We can use the information given in the problem to find the value of k. We know that an 8-inch candle burns down in 4 hours, so we can substitute these values into the equation above to find the value of k, as follows:
4 = k * 8
Solving for k, we get:
k = 0.5
Now that we know the value of k, we can use the equation above to determine how long it would take a 12-inch candle to burn down. Substituting the values for t, k, and l into the equation, we get:
t = (0.5) * (12) = 6
This means that it would take a 12-inch candle 6 hours to burn down.
Daliyah is given one point on a line as (3,−1) and the slope of the line as −5.
The y-intercept of the line having point (3,−1) and slope -5 is 14.
The equation of a line in the slope-intercept form is:-
y = mx+c
where 'm' is the slope and 'c' is the y-intercept.
here, m= -5
∴ y = -5x + c .........(Partial equation)
to find b substitute (3,-1) into the partial equation
where,
x= 3 and y= -1 by substituting the values,
-1 = (-5)×3 + c => c = 15 - 1
∵ c = 14
Thus, the value of the y-intercept is 14.
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The correct question is:-
Daliyah is given one point on a line as (3,−1) and the slope of the line as −5. what is the y-intercept of the line?
Help me with this question please
The plane x + y + z = 1 cuts the cylinder x2 + y2 = 1 in an ellipse. We want to find the points on the ellipse that lie closest to the origin. Using Lagrange Multiplier method, set up a system of equations for such points.
The points on the ellipse that lie closest to the origin are (1, 0, 0), (0, 1, 0)
Let (x, y, z) be a point on an ellipse in which plane x + y + z = 1 cuts the cylinder [tex]$x^2+y^2=1$[/tex]
Let f(x, y, z) = [tex]$x^2+y^2+z^2$[/tex]
[tex]& g_1(x, y, z)=x^2+y^2-1=0 \\[/tex]
[tex]& g_2(x, y, z)=x+y+z-1=0[/tex]
From Lagrange Multiplier Method.
There are multiple ways to define the method of Lagrange multipliers.
Suppose f and g are two functions such that they both have continuous partial derivatives.
Let (x0, y0, z0) ∈ S := {(x, y, z) : g(x, y, z) = 0} and ∇g(x0, y0, z0) ≠ 0.
The method of Lagrange’s multipliers is an important technique applied to determine the local maxima and minima of a function of the form f (x, y, z) subject to equality constraints
[tex]& \nabla f=\lambda \nabla y_1+\mu \nabla g_2 \\[/tex]
[tex]\Rightarrow & \langle 2 x, 2 y, 2 z\rangle=\lambda\langle 2 x, 2 y, 0\rangle+\mu\langle 1,1,1\rangle \\[/tex]
[tex]\Rightarrow & \langle 2 x, 2 y, 2 z\rangle=\langle 2 x \lambda+\mu, 2 y \lambda+\mu, \mu\rangle[/tex]
[tex]2 x & =2 x \lambda+\mu-(1) \\[/tex]
[tex]2 y & =2 y \lambda+\mu \ldots(2) \\[/tex]
[tex]2 z & =\mu \quad \ldots(3)[/tex]
From equation (1) [tex]$2 x(1-\lambda)=\mu=2 z$[/tex]
From equation (2) [tex]$2 y(1-\lambda)=\mu=2 z$[/tex]
Therefore if [tex]\lambda=1$[/tex], then z = 0
if [tex]\lambda \neq 1, \quad x=y=\frac{2}{1-\lambda}[/tex]
if z = 0, then after solving.
[tex]x^2+y^2=1$\\ $2 x+y=1$[/tex]
We get two points: [tex]${(1,0,0)} \geq {(0,1,0)}$[/tex]
and if x = y.
Then:
[tex]& x^2+y^2=1 \Rightarrow x^2+x^2=1 \Rightarrow 2 x^2=1 \Rightarrow x=\pm \sqrt{\frac{1}{2}} \\ & x+y+z=1[/tex]
x + y + z = 1
[tex]\Rightarrow[/tex] x + x + z = 1
[tex]\Rightarrow[/tex] 2 = 1 - 2x
= [tex]1-2\left(\pm \sqrt{\frac{1}{2}}\right)=1 \pm \sqrt{2}[/tex]
Therefore the corresponding points on the ellipse.
[tex]P_1\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 1-\sqrt{2}\right) \& P_2\left(-\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 1+\sqrt{2}\right)[/tex]
Now we have four points
Closest to the origin:
1) (1, 0, 0)
2) (0, 1, 0)
The farthest to origin:
3) [tex]$\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 1-\sqrt{2}\right)[/tex]
4) [tex]$\left(-\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 1+\sqrt{2}\right)[/tex]
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If BA BD, the value of x is:
4.
9.
10.
None of these choices are correct.
Nadia typically type 32 word per minute, but her rate may vary by a much a 5 word per minute. Let x be the actual rate at which Nadia type. What i the range of rate, in word per minute, that Nadia could type?
The range of rate that Nadia can type is 27 to 37.
We will calculate the lower and upper limit stating the minimum and maximum number of words Nadia can type in a minute. It will depict the range of rate.
Minimum number of words = 32 - 5
Performing subtraction on Right Hand Side of the equation
Minimum number of words = 27
Maximum number of words = 32 + 5
Performing subtraction on Right Hand Side of the equation
Minimum number of words = 37
Hence, Nadia's range of word typing is 27 to 37 per minute.
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Anjum and Betty spent a total of 9 hours completing the puzzle.
The ratio Anjum’s time : Betty’s time = 7 : 5.
Work out how much time Anjum spent on the puzzle.
please help me with question no. 5
If possible then answer with an explanation would be appreciated.
Answer:
a)
Step-by-step explanation:
additions do not work in general that way when we consider a mixture of positive and negative numbers.
e.g.
x = 3, y = -5
|3 + -5| >= |3| + |-5|
|-2| >= 3 + 5
2 >= 8
is definitely not true.
so, for positive/negative number mixtures a) is wrong.
b) works because
if x >= y, then |x - y| >= |x| - |y|. equal when x and y > 0. larger when x and/or y < 0. correct
if x < y, then |x - y| >= |x| - |y|. larger when x and y > 0.
equal when x and/or y < 0.
c) and d) work because the signs don't change anything. either the result is positive anyway, or if negative just gets converted to the positive result.
What is the third quartile of the given data set?
(2,5,6,7,15, 16, 17)
Answer:
16 is the third quartile
Find the coordinates of h if x(11, -6) is the midpoint of gh and g(15, 4).
Answer:
h (7, - 16 )
Step-by-step explanation:
given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint M is
M = ( [tex]\frac{x_{1}+x_{2} }{2}[/tex] , [tex]\frac{y_{1}+y_{2} }{2}[/tex] ) ← midpoint formula
let the coordinates of h be (x, y )
here (x₁, y₁ ) = g (15, 4 ) and (x₂, y₂ ) = h (x, y )
use the midpoint formula on g, h and equate to the coordinates x (11, - 6 )
[tex]\frac{15+x}{2}[/tex] = 11 ( multiply both sides by 2 )
15 + x = 22 ( subtract 15 from both sides )
x = 7
and
[tex]\frac{4+y}{2}[/tex] = - 6 ( multiply both sides by 2 )
4 + y = - 12 ( subtract 4 from both sides )
y = - 16
then
coordinates of h = (7, - 16 )
The rectangular floor of a classroom is 28 feet in length and 38 feet in width. A scale drawing of the floor has a length of 14 inches. What is the area, in square inches, of the floor in the scale drawing?
The area of the floor of the scale drawing and the scale factor are 72 in² and 1/2 respectively.
Actual Length = 28 feets
Actual width = 38 feets
Model width = 14 inches
The scale factor = 14/28 =1/2
The length of the model can be calculated thus :
Actual Length × scale factor
Model width = 38 × 1/2= 19 inches
The Area of a rectangle can be calculated using the relation :
Area = Length × width
Area of floor = 19 inches × 14 inches = 266 inch²
Therefore, the area of the floor of the scale drawing is 266 in²
What is area?Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.To learn more about area refer to:
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Write the log equation as an exponential equation. You do not need to solve for x.
log_5x(2x-4)=3x+2
If the value of the base is y, then y^(3x+2) = 5x(2x-4)
What is logarithm?logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if b^x= n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log8 base 2.
Also if 10² = 100 , then 2 = log100 base 10.
representing the unknown base to be y
then, log 5x(2x-4) base y = 3x+2
from the law of logarithm
log a base b = x, then b = a^x
Similarly log 5x(2x-4) base y = 3x+2
then 5x(2x-4) = y^(3x+2)
Therefore the exponential equation of the logarithm is 5x(2x-4) = y^(3x+2)
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(q014) how many of the twenty-five top-grossing films of 2017 in the united states were part of a movie franchise?
five out of of the twenty-five top-grossing films of 2017 in the united states were part of a movie franchise
If you were to attribute this film, that means you'd say that (something) caused the success of the movie Rocky. This question is basically asking what do you think made this movie a big hit? Personally, I'd say it's because of how the storyline and plot turned out. It's very intriguing and it doesn't just make the movie about boxing. The film adds some love between the protagonist and some other character, and not only that, but the movie teaches you something. Because Rocky lost to Apollo, Rocky didn't care. At least he still had Adrian. So, the movie teaches you that, even if you do lose, at least you still have something and/or someone with you.
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Solve each problem. Explain or show your reasoning.
a. What is 25% of 160?
b. What is 39% of 200?
c. What is 150% of 32?
d. 13 is 50% of what number?
e. 18 is 120% of what number?
f. 21 is what percentage of 30?
Answer:
Step-by-step explanation:
a. 40 because 25 percent is 1/4th and 160 times 1/4=40
b. 78 because 200 times 39/100 is 78
c. 48 because 32 times 150/100=48
d. 26 because if it is 50 percent it is 1/2 of a number so just multiply
e. 15
f. 70 percent, simplify both numbers to the lowest number possible equal to 21/30 so 7/10 and that is 70 percent
f ''(x) = x−2, x > 0, f(1) = 0, f(8) = 0
The required function f when f"(x) = x^-2, x > 0, f(1) = 0, f(2) = 0 is f(x) = - ln x + (ln 8) x/7 + (ln 8)/7
In tthis question we have been given the second derivative of function
f"(x) = x^-2, x > 0, f(1) = 0, f(2) = 0
We need to find a function f(x).
Consider f"(x) = x^-2
Integrating above expression with respect to x we get,
f'(x) = -1/x + C
againg integrating above expression with respect to x we get,
f(x) = - ln x + Cx + D
Now we need to find C and D.
Given that f(1) = 0
- ln (1) + C + D = 0
C + D = 0 ........... (1)
And f(8) = 0
- ln 8 + 8C + D = 0
8C + D = ln 8 ..............(2)
subtracting equation (1) from equation (2),
C = (ln 8)/7
Subtituting above value of C in equation (1) we get,
D = -(ln 8)/7
So, f(x) = - ln x + (ln 8) x/7 + (ln 8)/7
Therefore, the required function : f(x) = - ln x + (ln 8) x/7 + (ln 8)/7
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. The flashlight shown below has no batteries. Instead, it is operated by squeezing and letting go of the handle
Inside the body of the flashlight are gears.
A Kinetic → electrical → light
8 Kinetic chemical → light
C. Chemical -kinetic light
D chemical- electrical → light
The sequence of the energy conversion is; Kinetic → electrical → light Option A
What is energy conversion?From the first law of thermodynamics, we know that energy can neither be created nor destroyed but it can be converted from one form to the other. We have been told that in this case, the flashlight shown below has no batteries. Instead, it is operated by squeezing and letting go of the handle.
The letting go of the handle is kinetic energy which would lead to electrical energy that is then converted into the visible light energy.
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what is the angle of rotation that maps point P to point Q?
Answer:225 degrees
Step-by-step explanation:
A ship sails 40 nautical miles (nm) due west from point A and then changes course and sails 20 nautical miles in a direction that is 54° west of due north. A diagram of the ship’s trip is shown
A. Find m
B. Find the length of b to the nearest hundredth
.
C. the length of d to the nearest hundredth.
D.the distance from point A to point C.
E. Find m
F. the length of c to the nearest nautical mile (nm).
Answer:
Look at pic 2 for the measurements. I rounded to the thousandth, because some required rounding to the hunderedth and others required to the nearest mile and others were unspecified.
Step-by-step explanation:
We should first draw a diagram. Since you said a diagram is shown, I will assume that the diagram looks something like pic 1. I do not know which variables correspond to which lengths, so I used random variables, but the diagram I drew should help you figure out which variables I used correspond to your actual problem.
If so, we can use our trigonometric functions and Pythagorean theorem to solve for all the lengths
We must solve for x and y in the diagram, and we can do so by using sine and cosine. We know the orange angle of the triangle whose legs are x and y is 36 degrees. We also know the hypotenuse of that triangle is of length 20 nm. Thus, [tex]sin(36) = \frac{x}{20}[/tex] and [tex]cos(36) = \frac{y}{20}[/tex].
We can multiply both sides of both of these functions by 20 to get: [tex]x = sin(36) * 20[/tex] and [tex]y = cos(36) * 20[/tex]. These are approximately x = 11.756 nautical miles and y = 16.180 nautical miles
We can solve for the length of a by adding 40 to y and getting a = 16.180 + 40 = 56.180 nautical miles
We can solve for the length of z by using the Pythagorean theorem and the leg lengths of a and x. We can do [tex]\sqrt{(40 + 20cos(36))^2 + (20sin(36))^2}[/tex] and get z = [tex]\sqrt{3294.427191}[/tex] = 57.397 nautical miles
Thus, we get the measurements in pic 2 as the distances you are asked to solve for.
The lengths are 56.180 nautical miles and 57.397 nautical miles
What is Pythagoras theorem?Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”. The sides of this triangle have been named Perpendicular, Base, and Hypotenuse.
Given that, A ship sails 40 nautical miles (nm) due west from point A and then changes course and sails 20 nautical miles in a direction that is 54° west of due north.
We can use our trigonometric functions and Pythagorean theorem to solve for all the lengths
We must solve for x and y in the diagram, and we can do so by using sine and cosine. Furthermore, we also know the hypotenuse of that triangle is of length 20 nm. Thus, and.
We can multiply both sides of both of these functions by 20 to get: and. These are approximately x = 11.756 nautical miles and y = 16.180 nautical miles.
We can solve for the length of a by adding 40 to y and getting a = 16.180 + 40 = 56.180 nautical miles
We can solve for the length of z by using the Pythagorean theorem and the leg lengths of a and x. Likewise, we can do √40+(20cos(36°)²)+(20sin36°)²) and get z = 57.397 nautical miles.
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find f. f ''(x) = x−2, x > 0, f(1) = 0, f(3) = 0
Answer:
Step-by-step explanation:
Here we have the second derivative of a function and we need to go back to the original function. This situation is inverse of taking derivative which is called antiderivative or integral. But because we have the second derivative, we have to integrate twice because the first integral will give me the first derivative then the second integral will give me the function itself.
1. Let's start taking the 1st integral of x-2 which is x²/2-2x+C1
2. Take the 2nd integral which is the integral of the 1st one
x³/6-2x²/2+C1x+C2
remember: the integral is the inverse of derivative, that means we have to add 1 to the power of x and divide by that power. I did that in steps 1 and 2.
3. After simplifying the second term, f(x)=x³/6-x²+C1x+C2 this is the function, but I have to find C1 and C2 the constants. We can use the two conditions given. the 1st one means that when x=1 , y=0
0=1/6-1+C1+C2 now combine like terms and move them to the left side of the equation C1+C2=5/6
the 2nd condition means that when x=3 , y=0
0=27/6-9+3C1+C2 after simplifying 3C1+C2=9/2
4. We got 2 equations with 2 variables C1 and C2, we can solve these system of equations. You can eliminate C2 and find C1 which is 11/6 then plug in C1 in any of the two equations to find C2, which is -1
5. Last step substitute C1 and C2 in the function from step 3 and you will get f(x)=x³/6-x²+11x/6-1
[tex]y = 2x + 7[/tex]
[tex]y = 3 - \frac{1}{2} x[/tex]
Is this parallel, perpendicular, or neither?
The box plot shows the times for sprinters on a track team.
A horizontal number line starting at 40 with tick marks every one unit up to 59. The values of 44, 48, 50.5, 53, and 56 are all marked by the box plot. The graph is titled Sprinters' Run Times, and the line is labeled Time in Seconds.
Which value does 50% of the data lie above, and what is it called?
48, the lower quartile
50.5, the median
56, the median
53, the upper quartile
The value that 50% of the data, as shown in the box plot, lie above and what it is called is: B. 50.5, the median
What is the Median of a Box Plot?A box plot that displays the data set has the value of the median indicated at the point where a vertical line divides the rectangular box.
The median can be referred to as the second quartile (Q2) or 50th percentile of the data set.
The median value shows that 50% of the data set is above it while 50% is below it also.
The box plot in the given image shows that the vertical line crosses the rectangular box at 50.5.
50.5 is therefore the median, and this means that 50% of the data lie above 50.5.
Therefore, the answer is:
B. 50.5, the median
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which sample will have a larger chi-square test statistic? (see if you can tell without doing any calculations.)
The sample with more observations and/or more variability in the data will generally have a larger chi-square test statistic.
Chi-square test statistic is a measure of how well a set of observed data fits a theoretical model. The larger the sample size and the more variability in the data, the more likely it is that the theoretical model won't fit the data perfectly. Therefore, the larger sample size and variability in the data will lead to a larger chi-square test statistic.
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Which of the equations listed below would be parallel to the line divon by y= -4x + 3
Answer: A
A. y=-4x-2. Two lines are parallel if they have the same slope. In this case, the line given by y = -4x + 3 has a slope of -4, so the equation of the line that is parallel to it must also have a slope of -4. The equation y = -4x - 2 has a slope of -4, so it is the equation of a line that is parallel to the line given by y = -4x + 3.
Find the 12th term of the following geometric sequence. 2, 8, 32, 128
Step-by-step explanation:
So 2 * 4 = 8
8 * 4 = 32
32 * 4 = 128
128 * 4 = 512
512*4=2048
2048*4=8192
8192*4=32768
32768*4=131072
131072*4=524288
524288*4=2097152
2097152*4=8388608
8388608*4=33554432 and that would probably be your final answer
Tanisha has 51 m of fencing to build a
three-sided fence around a rectangular
plot of land that sits on a riverbank. (The
fourth side of the enclosure would be the
river.) The area of the land is 304 square
meters. List each set of possible
dimensions (length and width) of the
field.
Possible dimensions #1:
meters by
meters.
Possible dimensions #2:
meters by
meters.
Answer:
Two sets of dimensions are possible:
L(m) W(m) Area(m^2) Fence (m) Perimeter
Set 1 19 16 304 51
Set 2 32 9.5 304 51
Step-by-step explanation:
Let's set L and W for the Length and Width of the perimeter. We know that:
Area = L*W
Area = 304 m^2
Perimeter = 51 m
Let's say that the side bounded by the river is the length side. No fence is required for that portion, so the fencing for the perimeter is given by:
Perimeter, P = L + 2W
P = 51 m
We can write: L+2W=51 m
Rearrange to isolate one of the two variables. Lets pick L:
L+2W=51
L=51-2W
L = (51-2W)
Now use this in the area equation:
304 m^2 = L*W
304 m^2 = (51-2W)*W
304 m^2 = (51-2W)*W
304 m^2 = 51W-2W^2
Arrange this in the standard form of a quadratice equation and solve:
-2W^2+51W-304 m^2 = 0
W = 16 and 9.5 meters
W = 16 m
Since L=51-2W
L=51-2*(16)
L = 51-32
L = 19 m
Area = L*W
L*W = 304 m^2
(19m)W = 304 m^2
W = 16 m
Check:
Does a length (L) of 19 m and a width (W) of 16 m give an area of 304 m^2 and a fencing perimeter of 51 m if one of the L sides is the river?
Area = (19 m)*(16 m) = 304 m^2 YES
Per. = 2*(16)+ 19 = 51 m YES
W = 9.5 m
Ding the same calculation for a width of 9.5 m also satisfies all the equations, as per above.
Both values of W are valid, so the twos sets of possible dimensions are:
L(m) W(m) Area(m^2) Fence (m) Perimeter
Set 1 19 16 304 51
Set 2 32 9.5 304 51