[tex]S[/tex] is a closed surface with interior [tex]R[/tex], so you can use the divergence theorem.
[tex]\vec F(x,y,z)=x\,\vec\imath+y\,\vec\jmath+9\,\vec k\implies\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(x)}{\partial x}+\dfrac{\partial(y)}{\partial y}+\dfrac{\partial(9)}{\partial z}=2[/tex]
By the divergence theorem, the flux of [tex]\vec F[/tex] across [tex]S[/tex] is given by the integral of [tex]\nabla\cdot\vec F[/tex] over [tex]R[/tex]:
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV[/tex]
Convert to cylindrical coordinates, setting
[tex]x=u\cos v[/tex]
[tex]y=y[/tex]
[tex]z=u\sin v[/tex]
The integral is then
[tex]\displaystyle2\int_{v=0}^{v=2\pi}\int_{u=0}^{u=1}\int_{y=0}^{y=8-u\cos v}u\,\mathrm dy\,\mathrm du\,\mathrm dv=\boxed{16\pi}[/tex]
The flux of a vector field across a surface is calculated using a surface integral, which involves integrating the dot product of the vector field and the differential area element over the surface. For closed surfaces, the outward orientation is used. Electric flux, the flux of the electric field across a surface, is mentioned as an example.
Explanation:The flux of a vector field across a surface can be calculated using a surface integral. For the given vector field F(x, y, z) = x i + y j + 9 k and the surface S defined as the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 8, the flux is evaluated by integrating F·dS over the surface S.
The orientation of the surface is important in this process. For a closed surface, the positive (outward) orientation is used. The normal vector at any point on the surface points from the inside to the outside, and this outward normal is used to compute the flux through a closed surface.
Conceptually, this is like breaking the surface up into infinitesimally small patches dA and summing up the contributions of the vector field F (in this case, x i + y j + 9 k) through each patch - this is referred to as electric flux in the context of electric fields. This calculation embodies the concept of electric flux, which is defined as the scalar product of the electric field and the area vector.
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Members at a yoga school pay $10 per class plus a one time $100 membership fee.non members pay $15 per class.how many classes would a member have to take to save money compared to taking classes as a non-member?
Show work Plz!!
Answer:
21 classes
Step-by-step explanation:
Let’s set up equations for both memebers and non-members!
Let c = classes
To find how many classes it would take to make the two things even, we set them equal.
10c+100 = 15c
Solving for c,
5c = 100
So c = 20.
Since you need 20 classes to break even, to save money as a member, you need 21 classes.
Help please............
Answer:
(9x -2)(9x +2)
Step-by-step explanation:
Each of the terms in the difference is a perfect square, so the "perfect square trick" applies. The factors are the sum and difference of the square roots of the given terms.
√(81x²) = 9x√4 = 281x² - 4 = (9x +2)(9x -2)
Find the slope of the line that passes through the points (2.1) and (-1.-1).
Answer: The slope is [tex]\frac{2}{3}[/tex]
Step-by-step explanation:
The slope can be calculated with the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Then, knowing that the line passes through the points (2,1) and (-1,-1), we can substitute the coordinates into the formula.
In this case:
[tex]y_2=-1\\y_1=1\\x_2=-1\\x_1=2[/tex]
Therefore, the slope of the line that passes through the points (2,1) and (-1,-1) is:
[tex]m=\frac{-1-1}{-1-2}\\\\m=\frac{-2}{-3}\\\\m=\frac{2}{3}[/tex]
Find the value of x in each case. Give reasons to justify your solutions!
Answer:
x = 11°
Step-by-step explanation:
The parallel lines suggest we look to the relationships involving angles and transversals. The angle marked 33° and ∠CAB are alternate interior angles, hence congruent:
∠CAB = 33°
5x is the measure of the external angle opposite that internal angle and angle 2x of ΔABC, so it is equal to their sum:
5x = 2x + 33°
3x = 33° . . . . . . . . . subtract 2x
x = 11° . . . . . . . . . . . divide by 3
Choose the correct graph for the equation y=2x+3
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
Factor and Simplify the expression sin^2x+sin^2xcot^2x
[tex]\bf \textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ sin^2(x)+sin^2(x)cot^2(x)\implies sin^2(x)+\underline{sin^2(x)}\cdot \cfrac{cos^2(x)}{\underline{sin^2(x)}} \\\\\\ sin^2(x)+cos^2(x)\implies 1[/tex]
In a concert band, the probability that a member is in the brass section is 0.50. The probability that a member plays trombone, given that he or she is in the brass section, is 0.24. What is the probability that a randomly selected band member is in the brass section and plays trombone?
Answer:
B. 0.12
Step-by-step explanation:
To obtain this probability, you need to multiply the two probabilities.. since it's comprised of two events: one that he's in the brass section, one that he plays trombone. The probably of him playing trombone only happens if he's in the brass section.
So, you have the possibility he's in the brass section: 0.50
The possibility he's playing trombone, if he's in the brass section: 0.24
P = 0.5 * 0.24 = 0.12
Answer:
B: 0.12
Step-by-step explanation:
ap3x
14. Simplify 3! (1 point) 2 5 3 6
Answer: the answer is 6
Step-by-step explanation: 3 times 2 is 6 then 6 times 1 is still 6
Answer: [tex]3!=6[/tex]
Step-by-step explanation:
The factorial function has the following symbol: "!" (As you can observe, it is an exclamation mark).
The factorial functions are applied to the numbers greater than zero. Then, the factorial of a positive integer "n" is representend as:
[tex]n![/tex]
Since you need to simplify 3!, you has to multiply all the positive integer numbers that are between the number 3 and and the number 1.
Therefore, for 3! you get the following result:
[tex]3!=3*2*1\\3!=6[/tex]
PLEASE HELP!!!!! TIMED!!! Will give brainliest!! 70 POINTS!!!!!
Matteo wants to write a statement that can be represented by the inequality P<9 Which describes the correct method to write a statement to match this inequality?
Use the inequality symbol to determine the relationship between the variable and the constant. Decide what the variable will represent. Decide what the constant will represent. Write a statement using the same relationship. An example is “Alva spent more than $9.”
Decide what the variable will represent. Decide what the constant will represent. Write a statement using the same relationship. An example is “Alva spent less than $9.” Use the inequality symbol to determine the relationship between the variable and the constant.
Decide what the variable will represent. Decide what the constant will represent. Use the inequality symbol to determine the relationship between the variable and the constant. Write a statement using the same relationship. An example is “Alva spent less than $9.”
Decide what the variable will represent. Decide what the constant will represent. Use the inequality symbol to determine the relationship between the variable and the constant. Write a statement using the same relationship. An example is “Alva spent more than $9.”
Answer:C
Step-by-step explanation:
The constant and the variable relationship is p<9.That means it is "Alva spent less than 9%"
Answer:
it is c
Step-by-step explanation:
What is the determinant of m= {5 8 -5 4} ? 20 40 60 80
Answer:
60
Step-by-step explanation:
We have been given the matrix;
[tex]\left[\begin{array}{ccc}5&8\\-5&4\end{array}\right][/tex]
For a 2-by-2 matrix, the determinant is calculated as;
( product of elements in the leading diagonal) - (product of elements in the other diagonal)
determinant = ( 5*4) - (8*-5)
= 20 - (-40) = 60
Answer:
c. 60
Step-by-step explanation
math
If $22,000 is deposited in an account paying 3.85% interest compounded continuously, use the continuously compounded interest formula , A=Pe^rt, to find the balance in the account after 11 years.
A. $1,519,356.93
B. $33,600.60
C. $33,416.25
D. $25,416.25
Answer:
B
Step-by-step explanation:
In the equation for interest compounding continuously, the A stands for the amount after the compounding is done, the P is the initial amount invested, the e is Euler's number, the r is the rate in decimal form, and the t is the time in years that the money is invested. Setting up our equation with the given values looks like this:
[tex]A=22,000e^{(.0385)(11)}[/tex]
Multiply the rate with the time to simplify a bit to
[tex]A=22,000e^{.4235}[/tex]
Raise e to the power of .4235 on your calculator (hit 2nd then the ln button to get your e) and get
[tex]A=22,000(1.527297754)[/tex]
Multiply out to get $33600.55, but rounding up gives you B as your answer.
The balance in the account after 11 years is $33,600.6. Thus, the correct option is B.
What is continuous compounding?Theoretically, long-term average interest means that interest is continuously earned on a current account as well as reinvested into the balance to increase future interest earnings.
The equation is given as,
[tex]\rm P=P_o \times e^{rt}[/tex]
The balance in the account after 11 years is calculated as,
[tex]\rm P = \$22,000 \times e^{0.0385\times 11}[/tex]
Simplify the equation, then we have
P = $22,000 x 1.52729
P = $33,600.6
Thus, the correct option is B.
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The function f(x) is the wait time for an amusement park ride where x is the number of people in line. What is the practical domain for the function f(x)?
all integers
all whole numbers
all real numbers
all positive integers
Answer:
The domain would be the set of all whole numbers.
Step-by-step explanation:
Integers are { ......-3, -2, -1, 0, 1, 2, 3....}
Whole numbers are {0, 1, 2, 3, ...... }
Real numbers are all numbers except imaginary number,
Positive integers are {1, 2, 3, 4, ....}
Given,
In the function f(x),
x represents the number of people in line,
We know that number of people can not be negative and it can be 0,
Thus, the possible value of x are 0, 1, 2, 3,......
Also, the domain of a function is the set of all possible value of input,
Since, x represent the input for the function f(x),
Thus, the domain of f(x) would be the set of all whole number.
Second option is correct.
PLEASE HELP ME WILL RATE AND MARK BRAINLIEST PLEASE I NEED TO PASS IMAGE ATTACHED
Answer:
x = 21
Step-by-step explanation:
From the rules of secants (and tangents), you know that ...
x^2 = 7·(7 +56)
x = √(7·63) = √441 = 21
_____
When secant lines intersect, the product of distances from the point of intersection to the two points on the circle is a constant. The tangent line is a degenerate case where the two points of intersection are the same point (hence the product of distances is x^2). For the other secant, the near distance is 7, and the far distance is 7+56 = 63.
Interestingly, this property holds whether the secants intersect inside the circle or outside (as here). That makes it easier to remember, since there's really only one rule, not three.
Evaluate: 54-75+81-(-27)+53
Hi the answer is 344
Answer:
140
Step-by-step explanation:
54-75=-21
-21+81=60
60-(-27)=87 or 60+27=87
87+53=140
What is Mean Absolute Deviation (MAD)? How do I do it? (Please have an example to show.)
Explanation:
It is the average (mean) of the absolute values of the differences between a set of numbers and their mean.
Example: consider the set {1, 2, 4}. The mean is computed in the usual way: the sum divided by the number of contributors —
mean = (1 + 2 + 4)/3 = 7/3 = 2 1/3
Then the deviations are ...
1 -2 1/3 = -1 1/3 . . . . the absolute value of this is 1 1/3
2 -2 1/3 = -1/3 . . . . . the absolute value of this is 1/3
4 -2 1/3 = 1 2/3 . . . . the absolute value of this is 1 2/3
The mean of these absolute values is their sum divided by the number of them:
(1 1/3 +1/3 +1 2/3)/3 = (3 1/3)/3 = 1 1/9
The MAD of {1, 2, 4} is 1 1/9.
_____
Your graphing calculator or spreadsheet program may have a function that will calculate this for you.
Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C cos(y) dx + x2 sin(y) dy C is the rectangle with vertices (0, 0), (5, 0), (5, 4), (0, 4)
By Green's theorem,
[tex]\displaystyle\int_C\cos y\,\mathrm dx+x^2\sin y\,\mathrm dy=\iint_D\left(\frac{\partial(x^2\sin y)}{\partial x}-\frac{\partial(\cos y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy[/tex]
where [tex]D[/tex] is the region with boundary [tex]C[/tex], so we have
[tex]\displaystyle\iint_D(2x+1)\sin y\,\mathrm dx\,\mathrm dy=\int_0^5\int_0^4(2x+1)\sin y\,\mathrm dy\,\mathrm dx=\boxed{60\sin^22}[/tex]
I need help with Precal asap !!!! I’ll mark u as brainliest, please if you don’t know the correct answer don’t write down.
Answer:
Equation 1: r = 4 +( 3 * cos theta )
Equation 2: r = sqrt ( 5² * sin(2 theta) )
Step-by-step explanation:
GRAPH 1:
The first graph is a dimpled limacon.
General equation for dimpled limacon:
r = a + b cos theta ∴ if dimple is along the x- axis
r = a + b sin theta ∴ if dimple is along the y-axis
y-intercept : { a, -a } = { 4, -4 } ∴ the points at which limacon intersects y-axis
Negative side of x-axis = ( a – b ) ⇒ 1
Positive side of x-axis = ( a + b ) ⇒ 7
Subtract the value of a from sum of a and b to find b:
b = 7 – 4 ⇒ 3
Equation1: r = 4 +( 3 * cos theta )
GRAPH 2:
The second graph is a lemniscates.
General equation for lemniscates is:
r² = a² cos(2theta) ∴ if petals of graph are on coordinate axis
r² = a² sin(2 theta) ∴ if petals of graph are not on coordinate axis
now, according to the graph:
a = 5 ⇒ a² = 25
angle of graph: cos2θ, simply divide 360° by 2:
[tex]\frac{360}{2}[/tex] ⇒ 180°
The petals cannot be on coordinate axis, we start from 45° and then the next petal will be on:
45° + 180° = 225°
Since the graph is not on the coordinate axis, so
r² = 5² sin(2 theta) ⇒ r = sqrt ( 5² * sin(2 theta) )
Equation 2: r = sqrt ( 5² * sin(2 theta) )
In triangle ABC, the side lengths are AB = 13, AC = 21, and BC = x. Write a compound inequality that represents the range of possible values for x.
Big fraction
Parentheses
Vertical bars
Square root
Root
Superscript (Ctrl+Up)
Subscript (Ctrl+Down)
Plus sign
Minus sign
Middle dot
Multiplication sign
Equals sign
Less-than sign
Greater-than sign
Less-than or equal to
Greater-than or equal to
Pi
Alpha
Beta
Epsilon
Theta
Lambda
Mu
Rho
Phi
Sine
Cosine
Tangent
Arcsine
Arccosine
Arctangent
Cosecant
Secant
Cotangent
Logarithm
Logarithm to base n
Natural logarithm
Bar accent
Right left arrow with under script
Right arrow with under script
Angle
Triangle
Parallel to
Perpendicular
Approximately equal to
Tilde operator
Degree sign
Intersection
Union
Summation with under and over scripts
Matrix with square brackets
The compound inequality representing the range of possible values for 'x' (side length BC) in the triangle ABC, given that side lengths AB=13 and AC=21, is 8 < x < 34.
Explanation:In the triangle ABC with side lengths, AB=13, AC=21, and BC=x, we use the triangle inequality theorem which states that the length of a side of a triangle is less than the sum of the lengths of the other two sides and more than the absolute difference between them. Applying this theorem to your specific situation, we get the compound inequality 8 < x < 34. This inequality represents the range of possible values for the side length x is BC.
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For triangle ABC with sides AB = 13, AC = 21, and BC = x, the range of possible values for x is[tex]\( 8 < x < 34 \)[/tex] based on the triangle inequality theorem.
In triangle ABC, the relationship among its side lengths is governed by the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
For the given triangle with side lengths AB = 13, AC = 21, and BC = x, we can express this relationship as a compound inequality. Let \[tex]( a \), \( b \), and \( c \)[/tex] represent the lengths of the sides of the triangle. According to the triangle inequality theorem, we have:
[tex]\[ a + b > c \][/tex]
Substitute the given values:
[tex]\[ 13 + x > 21 \][/tex]
Now, solve for[tex]\( x \)[/tex]
[tex]\[ x > 8 \][/tex]
Similarly, for the other pair of sides:
[tex]\[ 21 + x > 13 \][/tex]
Solving for \( x \):
[tex]\[ x > -8 \][/tex]
However, since side lengths cannot be negative, we disregard the second inequality. Therefore, the compound inequality representing the range of possible values for [tex]\( x \) is \( 8 < x < 34 \).[/tex] This means that any value of [tex]\( x \)[/tex] between 8 and 34 (exclusive) will satisfy the conditions for the given triangle.
i'm gonna need help with this one
Answer:
KM = 20
Step-by-step explanation:
Point V is the midpoint of KM, so ...
KV = VM
2.5z = 5z -10
10 + 2.5z = 5z . . . . . add 10
10 = 2.5z . . . . . . . . . subtract 2.5z
This is sufficient to answer the question:
KV = VM = 2.5z = 10
KM = KV + VM = 10 + 10
KM = 20
_____
In this case, it is not necessary to find the value of z. If you wanted to, you could divide by the coefficient of z in the last equation:
10/2.5 = z = 4
2nd term in expansion of the binomial theorem (4x+2y^3)^3 show work
I hope this helps with you
PLEASE ANSWER THIS... WILL VOTE FOR U
The answer is g(x) = (1/4 x)^2
It has a horizontal stretch of 4. Since it is horizontal it goes inside the parentheses and becomes the reciprocal of 4 which is 1/4
Hope this helped!
~Just a girl in love with Shawn Mendes
Given: m EH =85°, m∠EYV=35°. Find: m EV .
Answer:
155°
Step-by-step explanation:
∠EYV is half the difference of arcs EV and EH
(1/2)(EV -EH) = ∠EYV
(1/2)(EV -85°) = 35° . . . . fill in the given values
EV -85° = 70° . . . . . . . . .multiply by 2
EV = 155° . . . . . . . . . . . . add 85°
Thaddeus and lan start at the same location and drive in opposite directions, but leave at different times. When they are 365 miles apart, their combined travel is 16 hours. If Thaddeus drives at a rate of 20 miles per hour and lan drives at a rate of 25 miles per hour, how long had each been driving?
Thaddeus has been driving____? hours and lan has been driving_____? hours.
I NEED HELP RIGHT NOW PLEASE
Answer:
Thaddeus: 7 hIan: 9 hStep-by-step explanation:
If Thaddeus drives the whole 16 hours, the distance between them is ...
distance = speed · time
distance = 20 mi/h · 16 h
distance = 320 miles.
It is 45 miles more than that. For each hour that Ian drives, their separation distance increases by (25 mph -20 mph)·(1 h) = 5 mi. Then Ian must have driven ...
(45 mi)/(5 mi/h) = 9 h
The rest of the 16 hours is the time that Thaddeus drove: 7 hours.
___
Let x represent the time Ian drives. Then 16-x is the time Thaddeus drives. Their total distance driven is ...
distance = speed · time
365 mi = (25 mi/h)(x) + (20 mi/h)(16 h -x)
45 mi = (5 mi/h)(x) . . . . . . . . subtract 320 miles, collect terms
(45 mi)/(5 mi/h) = x = 9 h . . . . . . divide by the coefficient of x
_____
Comment on the solution
You may notice a similarity between the solution of this equation and the verbal discussion above. (That is intentional.) It works well to let a variable represent the amount of the highest contributor. Here, that is Ian's time, since he is driving at the fastest speed.
To solve the problem, we need to set up two equations based on the information given in the problem. Solving the equations simultaneously gives Thaddeus has been driving for 7 hours and Ian has been driving for 9 hours.
Explanation:First, we'll define the variables. Let's say the time Thaddeus has been driving is T hours, and the time Ian has been driving is I hours. The total distance they covered is the sum of the distances each one traveled, which is 365 miles. Thaddeus travels at a rate of 20 mph, while Ian travels at 25 mph. This is represented by the equation 20T + 25I = 365.
Next, we know that the total time they have been driving is 16 hours, which gives us another equation, T + I = 16. Now we have a system of linear equations that we can solve simultaneously to find the values of T and I. The solution gives T = 7 hours and I = 9 hours. Hence, Thaddeus has been driving for 7 hours and Ian has been driving for 9 hours.
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The maximum grade allowed between two stations in a rapid-transit rail system is 3.5%. Between station A and station B, which are 260260 ft apart, the tracks rise 7 and one half7
1
2 ft. What is the grade of the tracks between these two stations? Round the answer to the nearest tenth of a percent. Does this grade meet the rapid-transit rail standards?
The grade of the tracks between station A and station B is nothing%.
(Type an integer or decimal rounded to the nearest tenth as needed.)
Answer:
2.9%
Step-by-step explanation:
The problem says the stations are 260 feet apart. Assuming that this is the horizontal distance between them, then the grade is:
7.5 / 260 × 100% = 2.9%
This is less than the maximum of 3.5%, so it meets the standards.
Given: JK tangent, KH=16, HE=12 Find: JK.
Check the picture below.
a weight lifter uses both 2 kg blocks as well as 5 kg blocks to set up his bar. he has a total of 30 blocks if the total weight of the bar is observed to be 120 kg then calculate the number of 5 kg blocks and 2 kg blocks
Answer:
The number of the 2 kg blocks is 10
The number of the 5 kg blocks is 20
Step-by-step explanation:
Let the number of the 2 kg blocks be x and the number of the 5 kg blocks be y.
The total number of blocks in terms of x and y is;
x + y
We are informed that he has a total of 30 blocks, implying that;
x + y = 30
The total weight of the bar in terms of x and y will be;
2x + 5y
We are also informed that the total weight of the bar is observed to be 120, implying that;
2x + 5y = 120
We therefore have two equations in two unknowns.;
x + y = 30
2x + 5y = 120
We can solve these equations simultaneously via elimination method;
multiply the first equation by 2;
2x + 2y = 60
2x + 5y = 120
Subtracting the first equation from the second one yields;
5y - 2y = 120 - 60
3y = 60
y = 20
But x + y = 30, implying that;
x = 30 - 20
x = 10
The perimeter of a rectangle is 72 centimeters. The base is 2 times the height. What are the dimensions of the rectangle? a. h = 9 cm, b = 18 cm c. h = 12 cm, b = 24 cm b. h = 19 cm, b = 38 cm d. h = 10 cm, b = 20 cm
The answer is c. This is the ONLY possible answer.
A car repair center services 920 cars in 2012. The number of cars serviced increases quarterly at a rate of 12% per year after 2012. Create an exponential expression to model the number of cars serviced after t years. Then, match each part of the exponential expression to what it represents in the context of the situation. The quarterly rate of growth is 0.03 or 3%. The growth rate is 1.03. The growth factor is represented by 1.03. The compound periods multiplied by the number of years is 4t. 920(1.03) is the number of cars multiplied by 1.03. The initial number of cars serviced is 920. Coefficient arrowRight Exponent arrowRight Rate arrowRight Base arrowRight
Answer:
N=920 (1+0.03)^4t
where N=number of cars serviced after t years
Step-by-step explanation:
Apply the compound interest equation
N=P( 1+r/n)^nt
where N ending number of cars serviced , P is the number of cars serviced in 2012, r is the interest rate, n is the number of compoundings per year, and t is the total number of years.
Matching parts of the exponential function
Initial number of cars serviced=920
The quarterly rate of growth = if interest is compounded quarterly, n=4
r=12% ÷ 4 = 0.03 or 3%
The growth rate is given by (1 +r/n) = 1+0.03 = 1.03
number of compoundings for t years= nt= 4t
The compound period multiplied by the number of years = 920(1.03)^4t
Solve for the indicated variable. Express numbers as integers or simplify fractions.
-288/5 or - 57.6 I used a calculator so I cannot solve step by step
A rectangle has an area of 12 square centimeters and a perimeter of 16 centimeters. Which of the following could be its dimensions? 2 cm and 6 cm 3 cm and 4 cm 1.5 cm and 8 cm 1 cm and 12 cm
Answer:
2 cm and 6 cm
Step-by-step explanation:
The product of the dimensions must be 12 cm². (All answer choices meet that requirement.)
Opposite sides of a rectangle are the same length, so the sum of the two dimensions must be half the perimeter, 8 cm. The sums of the answer choices are ...
8 cm7 cm9.5 cm13 cmOnly the first answer choice meets the requirement for a perimeter of 16 cm. The dimensions could be 2 cm and 6 cm.
6
×
2
Explanation:
For the rectangle
Length
=
ℓ
Breadth
=
b
Area is
12
cm
2
ℓ
b
=
12
Perimeter is
16
cm
2
(
ℓ
+
b
)
=
16
ℓ
+
b
=
8
Substitute
b
=
12
ℓ
from first equation
ℓ
+
12
ℓ
=
8
ℓ
2
+
12
=
8
ℓ
ℓ
2
−
8
ℓ
+
12
=
0
Use quadratic formula (
x
=
−
b
±
√
b
2
−
4
a
c
2
a
) to find
ℓ
ℓ
=
−
(
−
8
)
±
√
(
−
8
)
2
−
(
4
×
1
×
12
)
2
×
1
ℓ
=
8
±
√
16
2
ℓ
=
8
±
4
2
ℓ
1
=
8
+
4
2
=
6
ℓ
2
=
8
−
4
2
=
2
If
ℓ
1
is taken as length then
ℓ
2
is the breadth of the rectangle.