Answer:
The function is 1/4x^2 + x - 8
Step-by-step explanation:
In Factor form the function is
a(x + 8)(x - 4) where a is a constant to be found.
= a( x^2 + 4x - 32) Note the coefficient of x is positive because the function has a minimum value.
The line of symmetry is x = (-8+4)/2 = -2.
So the minimum value will be the value of f(x) when x = -2, so we have the equation
a(-2+8)(-2-4) = -9
-36a = -9
a = 1/4.
A company rents out 17 food booths and 26 game booths at the county fair. The fee for a food booth is $200 plus $5 per day. The fee for a game booth is $95 plus $10 per day. The fair lasts for d days, and all the booths are rented for the entire time. Enter a simplified expression for the amount, in dollars, that the company is paid
The expression that represents the total paid by the company is 295 + 15d.
What is a linear function?A straight line on the coordinate plane is represented by a linear function.
A linear function always has the same and constant slope.
In another word, a linear function is a function that varies linearly with respect to the changing variable.
As per the given,
The fee for a food booth is $200 plus $5 per day.
Total cost for d days will be = 200 + 5d
The fee for a game booth is $95 plus $10 per day.
Total cost for d days will be = 95 + 10d
Total cost for d days = (200 + 5d) + (95 + 10d)
⇒ 200 + 95 + 5d + 10d
⇒ 295 + 15d
Hence "The expression that represents the total paid by the company is 295 + 15d".
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To arrive to his appointment on time, Mr. Jones had to drive all the way from his home with the average speed of 60 mph. Due to heavy traffic, he was driving 15 mph slower than he planned and arrived to the appointment 20 minutes later. How many miles from Mr. Jones' home was his appointment?
Answer:
60
Step-by-step explanation:
Mr Jone's home distance is 60 miles away from his appointment place .
Distance = Speed x Time
Distance planned & distance covered is same.
Time covered is 20 minutes more than time planned ; Actual Speed is 15mph lower than planned speed (60 mph) , ie = 45 mph.
Let the planned time be = t , Actual time = t + 20 mints = t + 20 / 60 = t + 1/3 = 4 t / 3
As distance is same : 60 t = 45 ( 4t / 3)
60t = 60t
t = 1 hour
Distance = Speed x Time :
60 x 1 = 45 ( 4/3) = 60 miles
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Hello!! This is a A-F question, so it's kind of long, but I will definitely be making it worth your time if you could help me out with it!
Thank you so much :)
(Reporting any wrong answers so please don't post them!!)
Answer:
Please, see the attached files.
Step-by-step explanation:
Please, see the attached files.
Thanks.
At a heavy duty snow shovel was marked down by $6.00. The local sport committee purchased 3 shovels. The sale did not go well and the store owner made a new sal price by taking 1/3 off the original price.The sport committee purchased 3 more shovels. The total cost of the purchase was $102.00
Use the information to write an equation to calculate the original price if the shovel.
Answer:
The original price was $26.40
Step-by-step explanation:
Set up the equation(s) for this situation. The total price was for six shovels, three of which were discounted by 33.3% from the initially marked-down price X:
$102 = 3 * (X-$6) + 3 * ((X-$6) * (2/3))
102 = 3X - 18 + 3 * (2/3) * X - 3 * (2/3) * 6
102 = 5X - 30
X = $26.40
In an effort to control vegetation overgrowth, 139 139 rabbits are released in an isolated area free of predators. After 2 2 years, it is estimated that the rabbit population has increased to 556 556 . Assuming exponential population growth, what will the population be after another 6 6 months? Round to the nearest rabbit.
Answer:
197
Step-by-step explanation:
Initial population of rabbit is 139
after 2 years , rabbit population is 556
For exponential growth use y=ab^x
where a is the initial population
x is the time period
b is the growth rate, y is the final population
a= 139 is already given
when x=2, the value of y = 557
plug in all the values in the formula and find out 'b'
[tex]y=ab^x[/tex]
[tex]557=139(b)^2[/tex]
Divide both sides by 139
[tex]\frac{557}{139} =b^2[/tex]
take square root on both sides
b=2.00180 and b=-2.00180
growth factor cannot be negative
So b= 2.0018
The equation y=ab^x becomes
[tex]y=139(2.0018)^x[/tex]
To find population after 6 months
1 year = 12 months
so 6 months = 0.5 years
we plug in 0.5 for x
[tex]y=139(2.0018)^{0.5}[/tex]
y= 196.66
so population after 6 months = 197
Evaluate the infinite sum
The sum converges to 1000.
The [tex]n[/tex]-th partial sum of the series is
[tex]S_n=\displaystyle\sum_{i=1}^n100\left(\dfrac9{10}\right)^{i-1}=100\left(1+\dfrac9{10}+\left(\dfrac9{10}\right)^2+\cdots+\left(\dfrac9{10}\right)^{n-1}\right)[/tex]
Then
[tex]\dfrac9{10}S_n=100\left(\dfrac9{10}+\left(\dfrac9{10}\right)^2+\left(\dfrac9{10}\right)^3+\cdots+\left(\dfrac9{10}\right)^n\right)[/tex]
so that
[tex]S_n-\dfrac9{10}S_n=\dfrac1{10}S_n=100\left(1-\left(\dfrac9{10}\right)^n\right)[/tex]
[tex]\implies S_n=1000\left(1-\left(\dfrac9{10}\right)^n\right)[/tex]
As [tex]n\to\infty[/tex], [tex]\left(\dfrac9{10}\right)^n\to0[/tex], so we're left with
[tex]\displaystyle\sum_{i=1}^\infty100\left(\dfrac9{10}\right)^{i-1}=\lim_{n\to\infty}S_n=1000[/tex]
The given infinite series is a converging geometric series with an initial term of 100 and a common ratio of 9/10. Using the formula for the sum of an infinite geometric series, we find that the sum is 1000.
Explanation:To evaluate an infinite sum, or a series, we need to recognize the series structure. The given series ∑^{∞}_{i=1} 100(9/10)^{i-1} is a geometric series where the initial term (a) is 100 and the common ratio (r) is 9/10.
A geometric series converges only when the absolute value of r is less than 1, which is true in this scenario. When it converges, the sum (S) of the infinite geometric series can be calculated using the formula S = a / (1 – r).
By plugging into this formula, we get: S = 100 / (1 - 9/10) = 100 / (1/10) = 1000.
Therefore, the sum of the infinite series ∑^{∞}_{i=1} 100(9/10)^{i-1} is 1000.
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what is the perimeter of triangle with side lengths of 29, 15, and 4xy?
The perimeter of any polygon is equal to the sum of the length of all sides of this polygon.
Therefore:
P = 29 + 15 + 4xy = 44 + 4xyAnswer:
The perimeter of a triangle is defined as the sum of all three sides.So, we know that sides are 29, 15 and 4xy long. The perimeter would be
[tex]P=29+15+4xy[/tex]
Now, we sum like terms
[tex]P=44+4xy[/tex]
Therefore, the perimeter of the triangle is[tex]P=44+4xy[/tex]
This is a very difficult question. In △PQR, point T is on side QR such that QT=6 and TR=10. What is the ratio of the area of △PQT to the area of △PTR. Help please
Answer:
the ratio of the area is 3:5
Step-by-step explanation:
Find a common factor of 6 and 10 and then divide it, the common factor is 2
6 divided by 2 is 3
10 divided by 2 is 5
so your ratio area is 3:5
Hope this helps :)
Find the 10th partial sum of the arithmetic sequence defined by
Answer:
22.5
Step-by-step explanation:
If you expand the series, you can see the first few terms of the series:
Putting 1 in [tex]n[/tex], [tex]\frac{1}{2}(1)-\frac{1}{2}=0[/tex]Putting 2 in [tex]n[/tex], [tex]\frac{1}{2}(2)-\frac{1}{2}=0.5[/tex]Putting 3 in [tex]n[/tex], [tex]\frac{1}{2}(3)-\frac{1}{2}=1[/tex] Putting 4 in [tex]n[/tex], [tex]\frac{1}{2}(4)-\frac{1}{2}=1.5[/tex]We can see the series is 0, 0.5, 1, 1.5, ....
This is an arithmetic series with common difference (the difference in 2 terms) 0.5 and first term 0.
We know formula for sum of arithmetic series:
[tex]s_{n}=\frac{n}{2}(2a+(n-1)d)[/tex]
Where,
[tex]S_{n}[/tex] denotes the nth partial sum[tex]a[/tex] is the first term (in our case it is 0)[tex]n[/tex] is the term (in our case it is 10 since we want to find 10th partial sum -- sum until first 10 terms)[tex]d[/tex] is the common difference (difference in term and the previous term) (in our case it is 0.5)Substituting these into the formula, we get the 10th partial sum to be:
[tex]s_{10}=\frac{10}{2}(2(0)+(10-1)(0.5))\\s_{10}=5(0+(9)(0.5))\\s_{10}=5(0+4.5)\\s_{10}=5(4.5)\\s_{10}=22.5[/tex]
So the sum of the first 10 terms is 22.5. Third answer choice is right.
Answer is 22.5 so c :)
what is the measure of each exterior angle of a regular octagon is ___ the measure of each exterior angle of a regular hexagon.
A- Greater than
B- Less than
C- Equal to
E = 360/n
is the formula to use when computing the exterior angle E for any regular polygon with n sides. For an octagon, we have 8 sides meaning n = 8 leads to
E = 360/n = 360/8 = 45
The exterior angle of a regular octagon is 45 degrees
Repeat for n = 6 (hexagon) to get E = 360/n = 360/6 = 60. A regular hexagon has exterior angles of 60 degrees each.
We see that the regular octagon's exterior angles (45) are smaller than the regular hexagon's exterior angles (60)
-------------------------------------------
Answer: less than (choice B)
Final answer:
The measure of each exterior angle of a regular octagon is less than the measure of each exterior angle of a regular hexagon since the sum of the exterior angles is always 360 degrees and there are more sides on an octagon to divide this sum. The correct option is: B- Less than
Explanation:
To determine whether the measure of each exterior angle of a regular octagon is greater than, less than, or equal to the measure of each exterior angle of a regular hexagon, we must first understand how to calculate the measure of an exterior angle in a regular polygon. The sum of the exterior angles of any polygon is always 360 degrees, regardless of the number of sides. Therefore, to find the measure of a single exterior angle, you would divide 360 degrees by the number of sides the polygon has.
For a regular hexagon, which has six sides, the exterior angle is calculated as 360 ÷ 6, which equals 60 degrees. For a regular octagon, which has eight sides, the exterior angle is calculated as 360 ÷ 8, which equals 45 degrees.
Comparing the two measurements, we can clearly see that the measure of each exterior angle of a regular octagon is less than the measure of each exterior angle of a regular hexagon.
If the mass of a material is 45 grams and the volume of the material is 11 cm^3, what would the density of the material be?
I need the answer and than units
Answer:
Density of material would be 4.09 [tex]g/cm^3[/tex]
units is [tex]g/cm^3[/tex]
Step-by-step explanation:
Given: The mass of a material is 45 grams and the volume of the material is 11 cubic centimeter
Density is defined as mass per unit volume.
It is given by:
[tex]p= \frac{m}{V}[/tex] where p is the density , m is the mass and V is the volume of the material respectively.
Here, Density is expressed in grams per centimeter cubed (g/cubic cm)
Here, m = 45 g , V = 11 cubic cm
We get;
[tex]p= \frac{45}{11}[/tex] = 4.09 [tex]g/cm^3[/tex]
therefore, density of a material would be, [tex]4.09 g/cm^3[/tex]
and its units is [tex]g/cm^3[/tex]
How much does a length of aluminum expand if it is 1.2 meters long at 15 degrees c and heated to 65 degrees c
IN AN ATHLETIC EVENT 60 STUDENTS OF A SCHOOL PARTICIPATED LAST YEAR . THIS YEARTHE NUMBER OF STUDENTS OF THAT SCHOOLK TAKING PART IS DECREASED BY 5 % . FIND THE NUMBER OF STUDENTS TAKING PART IN THE ATHLETIC EVENT THIS YEAR
Answer: the number of students participating this year is 57.
Step-by-step explanation:
5% of 60 is 3.
60-3=57
I believe this is correct :)
Answer: Number of students participated this year = 57
Step-by-step explanation: Number of student participated last year = 60
Decreased by 5%
Decreased in number = 5% of 60 = 0.05x 60 =3
Number of Students this year = Number of student last year - decreased
= 60 - 3
= 57
the linear regression equation for a data set is y=-2.8x+70.8 where y is the temperature in degrees fahrenheit and x is the number of hours since 8am.what does the slope of the equation represent?
Step-by-step explanation:
We have been given the linear regression equation for a data set is y=-2.8x+70.8 where y is the temperature in degrees Fahrenheit and x is the number of hours since 8 am.
We can see that our given line is in slope-intercept form: [tex]y=mx+b[/tex], where, m = Slope of line and b = y-intercept.
Upon comparing our given equation with slope intercept form we can see that the slope of the equation (m) is -2.8. The slope represents that temperature is dropping at a rate of 2.8 Fahrenheit per hour after 8 am.
Given that AD and BC bisect each other at E, which of the following justifies ΔABE ≅ ΔDCE? A. Definition of Segment Bisector B. SSS Postulate C. Definition of Congruent Triangles D. SAS Postulate
Answer:
A. Definition of Segment Bisector
Step-by-step explanation:
One have to understand that according to data given in the question, we only know that AD and BC are bisected at the intersection point E. Now two triangles are formed which are ΔABE and ΔCDE.
Now by definition of segment bisector, we know that
AE = DE
BE = CE
Now, what is to understand that this information is based on the clue which is given in the question that AD and BC bisects each other. All the remaining options like SAS postulate, SSS postulate and definition of congruent triangle are not useful here if we don't know that these two lines bisect each other. Because, the fact that
AE = DE
BE = CE
is only derived by the information that AD and BC bisect each other. Now we can derive SSS and SAS postulate both because we know by the theorems of trigonometry that if two sides of two different triangles are equal in length, then their third sides must be equal, or when two lines bisect or intersect each other, vertical angles are always equal. So the answer is A.
What binomial do you have to add to the polynomial x^2+y^2–2xy+1 to get a polynomial: not containing the variable x
Pls Help me!
Answer:
Add [tex]-x^2+2xy[/tex]
Step-by-step explanation:
The polynomial [tex]x^2+y^2-2xy+1[/tex] can be added to eliminate the x terms by adding the additive inverse. We add [tex]-x^2+2xy[/tex] which has the inverse sign value of the polynomial terms.
[tex](x^2+y^2-2xy+1)+(-x^2+2xy)[/tex]
[tex]x^2-x^2+y^2-2xy+2xy+1[/tex]
When we simplify, this leaves [tex]y^2+1[/tex] without an x term.
Answer:
-x^2+2xy
Step-by-step explanation:
x^2 + y^2 -2xy + 1 +something = y^2 +1
This will get rid of the x and x^2 terms
Subtract y^2 from each side
x^2 + y^2 -y^2 -2xy + 1 +something = y^2-y^2 +1
x^2 -2xy+1 +something = 1
Subtract 1 from each side
x^2 -2xy+1 -1+something = 1-1
x^2 -2xy+something = 0
Subtract x^2 from each side
x^2 -x^2 -2xy+something = 0-x^2
-2xy+something = -x^2
Add 2xy to each side
2xy -2xy+something = -x^2+2xy
something = -x^2+2xy
We need to add -x^2+2xy
Remember a binomial is 2 terms
please help on this ? :)
Which inequality is represented by this graph? Also, if someone could tell me how this works that would be great :D
Answer:
y > -1/6 x+1
Step-by-step explanation:
The graph line is dotted. Dotted lines means greater than or less than. ( <, >)
Solid lines means less than or equal to or greater than or equal to (≤,≥)
Since the graph is shaded above the line, that means y is greater than the line. If it were shaded below the line, y would be less than.
y > -1/6 x+1
I WILL GIVE BRAINLIEST!!!!
The equation tells you that Henry swims 1.6·1 = 1.6 laps when x = 1 minute.
The table tells you Larry swims 4.5 laps in 2.5 minutes. Dividing these numbers by 2.5 tells you Larry swims 4.5/2.5 = 1.8 laps in 2.5/2.5 = 1 minute.
Henry's rate is 1.6 laps per minute; Larry's rate is 1.8 laps per minute.
___
1.8 is larger than 1.6, so Larry swims faster than Henry. That is, Larry swims farther in the same amount of time, or takes less time to swim the same distance.
It's costs $35 per hour to rent a boat at the lake you also need to pay a $25 fee for safety equipment you have $200 for how long can you rent the boat
Answer: 5 hours
Step-by-step explanation:
$35 per hour is the rate
$25 is the flat fee
$200 is the maximum you can spend
⇒ 35x + 25 ≤ 200
-25 -25
35x ≤ 175
÷35 ÷35
x ≤ 5
Solve for x 15x+5x 14x-6
Look at the picture.
[tex\alpha+\beta=180^o[/tex] - supplementary angles
Therefore we hve the equation:
[tex](14x-6)+(15+5x)=180\\\\(14x+5x)+(-6+15)=180\\\\19x+9=180\qquad\text{subtract 9 from both sides}\\\\19x=171\qquad\text{divide both sides by 19}\\\\\boxed{x=9}[/tex]
At a company fish fry, 1/2 in attendance are employees. Employees’ spouses are 1/3 of the attendance. What is the percentage of the people in attendance who are NOT employees or employee spouses?
SHOW WORK
Answer:At a company fish fry, 1/2 in attendance are employees.
Employees’ spouses are 1/3 of the attendance. What is
the percentage of the people in attendance who are not
employees or employee's spouses?
One half of X are employees.
One third of X are employee's spouses.
Step-by-step explanation:1/2 + 1/3 = 3/6 + 2/6 = 5/6
Then we subtract that fraction from one whole, or 1,
to see what fraction is left. That is, we say
1 minus 5/6
which is
1 - 5/6
Now write 1 as 6/6
6/6 - 5/6
We get 1/6
So 1/6 is left. Now we need to make that into
a percent by multiplying it by 100 and tacking on a "%"
1/6 × 100
1/6 × 100/1
100/6
50/3
16 2/3 %
So 16 2/3 % of the people in
attendance are neither employees
nor employee's spouses.
Now let's check:
1/2 are employees. That's 50%
1/3 are wmployee's spouses. That's 33 1/3%
15 2/3 % are neither employees nor employee's spouses.
Add them up
50 %
33 1/3 %
16 2/3 %
--------
99 3/3 %
And 99 3/3% = 100%
16.67% of the people in attendance are neither employees nor their spouses.
To calculate the percentage of people who are neither employees nor their spouses, we need to consider the total attendance as 100%. Since half of the attendance, i.e., 50%, are employees and one third, i.e., about 33.33%, are spouses, we add these figures to find the combined percentage of employees and spouses. We then subtract this combined percentage from 100% to find the percentage of people who are neither.
Combined percentage of employees and spouses: 50% (employees) + 33.33% (spouses) = 83.33%
Now, to find the percentage who are neither employees nor spouses, we subtract the combined percentage from 100%:
Percentage of neither: 100% - 83.33% = 16.67%
Therefore, 16.67% of the people in attendance are neither employees nor their spouses.
If ?ABC and ?XYZ are similar, which must be true? A) BC YZ = AC YX B) BC YZ = BA XZ C) AC XZ = BC YZ D) AC XZ = BA XZ
Answer:
The correct option is C.
Step-by-step explanation:
Let triangle ABC and XYZ are similar.
If two triangles are similar, then ratio of their corresponding sides are same.
Since ABC and XYZ are similar, therefore sides AB, BC and AC are corresponding to the sides XY, YZ and XZ respectively.
[tex]\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}[/tex]
From the above equation we can conclude that
[tex]\frac{BC}{YZ}\neq \frac{AC}{YX}[/tex]
[tex]\frac{BC}{YZ}\neq \frac{BA}{XZ}[/tex]
[tex]\frac{AC}{XZ}\neq \frac{BA}{XZ}[/tex]
Therefore option A,B and D are incorrect.
Answer: BC/YZ = AC/XZ
Step-by-step explanation:
Remember the end letters need to be the same. So if you look at your choices this is the only one that is in order BC AC
YZ XZ
took the test
Hendrick wants to enlarge a photo that is 4 inches wide and 6 inches tall. The enlarged photo keeps the same ratio. How y'all is the enlarged photo if it is 12 inches wide?
Answer:
Step-by-step explanation:
Alright, lets get started.
The original photo size is 4 inches wide and 6 inches tall.
So, the ratio of width and height will be = [tex]\frac{4}{6}=\frac{2}{3}[/tex]
The new enlarged photo will be of the same ratio means 2:3
The width of enlarged photo is given as 12 inches.
Suppose new height of enlarged photo is H, so
[tex]\frac{12}{H}=\frac{2}{3}[/tex]
Cross multiplying
[tex]2H=36[/tex]
Dividing 2 in both sides
[tex]H=18[/tex] inches
So the height of new enlagred photo will be 18 inches. : Answer
Hope it will help :)
for what values of k does kx^(2)-3x+2=0 have two equal real roots?
Tienes que usar la fórmula cuadrática:
(-b +/- √(b^2-4ac))/2a
Primero identificas los valores de a,b y c en kx^2-3x+2=0
K=a, b=-3, c=2
Luego sustituis en la fórmula y te queda:
(3+/-√(9-8k))/2k
Para que las raíces Sena reales se tienen que cumplir que 9-8k>=0
Answer:
Step-by-step explanation:
Therefore discriminant = b^2-4ac =0
b=-k, a=3,c=2
b^2-4ac= (-k)^2-4*3*2=0
k^2-24=0
k^2=24
k= +/- 2sqrt(6)
Which of the following best represents the relationship between angles A and B?
A = B
A = 180 degrees − B
B = 180 degrees − A
A = 2B
Answer:
A = B
Step-by-step explanation:
This is because they are alternate exterior angles and they equal the same thing.
if 3y - 7= 23, then y =
Answer:
y=10 is the value of y
Step-by-step explanation:
The given equation is
3y - 7 =23 ..............................(i)
We have to find out the value of y from the equation (i)
Now the equation is
3y - 7 = 23
adding 7 on both sides of the equation
3y - 7 + 7 = 23 + 7
3y = 30
as we need the value of y so
Dividing both sides of the equation by 3
[tex]\frac{3y}{3}=\frac{30}{3}[/tex]
which will lead us to
y = 10
so this is the value of y
Answer:
y=10
Step-by-step explanation:
3y-7 =23
3y = 23+7 -move 7 over
3y=30 -add remaing
3y/3 =30/3 -divide 3
y=10 -answer
check work 3x10-7=23
Brenda drove 3times as far as Jan Brenda drove 24 more miles than Jan how far did Jan drive
Identify the equation in point-slope form for the perpendicular bisector of the segment with endpoints B(−1,1) and C(−5,−7). PLEASE HELP!!!
Answer:
Equation in point-slope form= [tex]{y+3}=\frac{-1}{2}(x+3)[/tex]
Step-by-step explanation:
The given end points are B(−1,1) and C(−5,−7)
Mid point M of BC= [tex]\frac{-5-1}{2}[/tex] , [tex]\frac{-7+1}{2}[/tex]
Mid point M of BC = -3 , -3
Slope of BC = [tex]\frac{-7-1}{-5+1}[/tex] = 2
Slope of bisector= m= [tex]\frac{-1}{2}[/tex]
Equation of perpendicular bisector : [tex]\frac{y+3}{x+3}=\frac{-1}{2}[/tex]
⇒ [tex]{y+3}=\frac{-1}{2}(x+3)[/tex]
⇒ 2(y+3)= -(x+3)
⇒ [tex]2y+x=-9[/tex]
guys please answer me soon with an easy explanation
the diagonal of a rectangle is 20 metre and its parameter is 50 metre then what are its dimensions?
Answer:
Length 19.11 and width 5.89.
Step-by-step explanation:
Let the length be x and width be y metres.
Then, using the Pythagoras theorem:-
x^2 + y^2 = 20^2 = 400....................(1)
The perimeter = 50 so:-
2x + 2y = 50
Dividing through by 2:-
x + y = 25 .............................(2)
So y = 25 - x
Substitute for y in equation (1):-
x^2 + (25 - x)^2 = 400
x^2 + 625 - 50x + x^2 = 400
2x^2 - 50x + 225 = 0
x = 19.11 , 5.89, x = 19.11 as its the length
and y = 25 - 19.11 = 5.89 ( from equation (2).
"Parameter" = Perimeter.
Look at the picture.
We have the perimeter = 50 m.
The perimeter is 2l + 2w (l - length, w - width). Therefore
2l + 2w = 50 divide both sides by 2
l + w = 25 subtract w from both sides
l = 25 - w.
Use the Pythagorean theorem:
[tex]l^2+w^2=20^2\to(25-w)^2+w^2=20^2[/tex]
Use (a - b)² = a² - 2ab + b²
[tex]25^2-2(25)(w)+w^2+w^2=400\\\\625-50w+2w^2=400\qquad\text{subtract 400 from both sides}\\\\225-50w+2w^2=0\\\\2w^2-50w+225=0[/tex]
Use quadratic formula:
[tex]ax^2+bx+c=0\\\\\Delta=b^2-4ac\\\\x_1=\dfrac{-b-\sqrt\Delta}{2a};\ x_2=\dfrac{-b+\sqrt\Delta}{2a}[/tex]
We have:
[tex]a=2,\ b=-50,\ c=225[/tex]
Substitute:
[tex]\Delta=(-50)^2-4(2)(225)=2500-1000=1500\\\\\sqrt\Delta=\sqrt{1500}=\sqrt{100\cdot15}=\sqrt{100}\cdot\sqrt{15}=10\sqrt{15}\\\\w_1=\dfrac{-(-50)-10\sqrt{15}}{2(2)}=\dfrac{50-10\sqrt{15}}{4}=\dfrac{25-5\sqrt{15}}{2}\\\\w_2=\dfrac{-(-50)+10\sqrt{15}}{2(2)}=\dfrac{50+10\sqrt{15}}{4}=\dfrac{25+5\sqrt{15}}{2}[/tex]
[tex]l_1=25-w_1\\\\l_1=25-\dfrac{25-5\sqrt{15}}{2}=\dfrac{50}{2}-\dfrac{25-5\sqrt{15}}{2}=\dfrac{50-25+5\sqrt{15}}{2}=\dfrac{25+5\sqrt{15}}{2}\\\\l_2=25-w_2\\\\l_2=25-\dfrac{25+5\sqrt{15}}{2}=\dfrac{50}{2}-\dfrac{25+5\sqrt{15}}{2}=\dfrac{50-25-5\sqrt{15}}{2}=\dfrac{25-5\sqrt{15}}{2}[/tex]
[tex]Answer:\ \boxed{\dfrac{25+5\sqrt{15}}{2}\ m\times\dfrac{25-5\sqrt{15}}{2}\ m}[/tex]