Rectangle R has varying length l and width w but a constant perimeter of 4 ft. A. Express the area A as a function of l. What do you know about this function? B. For what values of l and w will the area of R be greatest? Give an algebraic argument. Give a geometric arguement.
Answers
l = length of the rectangle
w = width of the rectangle
P = 4 ft, constant perimeter
Because the given perimeter is constant,
2(w + l) = 4
w + l = 2
w = 2 - l (1)
Part A.
The area is
A = w*l
= (2 - l)*l
A = 2l - l²
This is a quadratic function or a parabola.
Part B.
Write the parabola in standard form.
A = -[l² - 2l]
= -[ (l -1)² - 1]
= -(l -1)² + 1
This is a parabola with vertex at (1, 1). Because the leading coefficient is negative the curve is downward, as shown below.
The maximum value occurs at the vertex, so the maximum value of A = 1.
From equation (1), obtain
w = 2 - l = 2 - 1 = 1.
The maximum value of the area occurs when w=1 and l=1 (a square).
Answer:
The area is maximum when l=1 and w=1.
The geometric argument is based on the vertex of the parabola denoting maximum area.
Related Questions
The diffrence between a term and
coefficient
Answers
- A coefficient tells you how many times to multiply a variable.
· Ex: 4xy = Coefficient: 4
- A term is each of the numbers in an equation, ratio, etc.
· Ex: 5x + 7y + (4x - 4y) + 2 = Terms: 5x, 7y, (4x - 4y), and 2
I am 99.9% sure this is the correct answer, but if it isn't I am truly sorry, and please forgive me.
150 centimeters is equivalent to
Answers
Which shows 54^2 − 46^2 being evaluated using the difference of squares method?
54^2 − 46^2 = (2916 + 2116)(2916 − 2116) = 4,025,600
54^2 − 46^2 = (54 + 46)(54 − 46) = (100)(8) = 800
54^2 − 46^2 = 2916 − 2116 = 800
54^2 − 46^2 = (54 − 46)^2 = 8^2 = 64
Answers
(a^2-b^2)=(a+b)(a-b)
So in this case:
(54^2-46^2)=(54+46)(54-46)=(100)(8)=800
hope it helps
Help asap plz ill give a gold medal. the label on cars antifreeze claims to protect the car between -30celsius and 130celsius. to convert Celsius temperature Fahrenheit temperature, the formula is, c=5/9(F-32). Write an solve and inequality to determine the Fahrenheit temperature range at which antifreeze protects the car.
Answers
[tex]-30\ \textless \ \frac{5}{9} (F-32)\ \textless \ 130[/tex]
Now it's just a matter of solving the inequalities simultaneously. Get rid of the fraction by multiplying everything by 9:
[tex]-270\ \textless \ 5(F-32)\ \textless \ 1170[/tex]
Then distribute the 5 into the parenthesis:
[tex]-270\ \textless \ 5F-160\ \textless \ 1170[/tex]
Now add 160 everywhere:
[tex]-110\ \textless \ 5F\ \textless \ 1330[/tex]
and finally divide everything by 5:
-22<F<266
What is the answer? (Tip- to undo multiply both sides by 4/7)
x|4/7 = 28
Answers
x / |4/7| = 28
Multiply by |4/7|
x = |4/7| x 28
Ignore the absolute for a second and note 4/7 x 28 is 16 because...
28 / 7 = 4
4 x 4 = 16
x = |16|
How to find the x intersept
Answers
For problem 22, the x intercept is the point (-4,0) as this is where the S shaped curve crosses through the horizontal x axis. See the attached diagram for a visual of what I'm talking about. Note the highlighted point in red.
last question
help me pls c:
Answers
(05.02)
What is the y-intercept of the line shown?
−1
0
0.5
1
Answers
A carpenter trims a triangular peak of a house with three 7-ft pieces of molding. The carpenter uses 21 ft of molding to trim a second triangular peak. Are the two triangles formed congruent? Explain.
Answers
The two triangles formed by trimming the peaks of the house with the 7-ft pieces of molding are congruent.
Explanation:To determine if the two triangles formed by trimming the peaks of the house with the 7-ft pieces of molding are congruent, we can use the concept of the Side-Angle-Side (SAS) congruence criterion.
In the first case, the carpenter uses three 7-ft pieces of molding to trim the first triangular peak. This means that each side of the triangle is 7 feet long.
In the second case, the carpenter uses 21 ft of molding to trim the second triangular peak. Since the total length of molding used is 21 ft, we know that each side of the triangle is still 7 feet long.
So, in both cases, the triangles are formed by sides of the same length, which is 7 feet, and they have a common angle at the peak of the house.
This satisfies the SAS congruence criterion, which states that if two triangles have two sides of equal length and the included angle is the same, then the triangles are congruent.
Therefore, the two triangles formed by trimming the peaks of the house with the 7-ft pieces of molding are congruent.
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What number is 7 units to the left of -1?
Answers
-1 - 7 = -8
answer
- 8
-1 - 7 = -8
Cost to rent a bicycle is $5 plus $3 per hour for x hours what equation represents the total cost for c hours
Answers
Hope I helped :)
The best approximation for the square root of 10 is.. A).5 B).100 C).3.1 D).25
Answers
Answer:
It is approximately 3.1
Step-by-step explanation:
The area of a rectangular plot 24 feet long and 16 feet wide will be doubled by adding an equal distance to each side of the plot. What is the distance added to each side?
Answers
24*16 = 384
384*2 = 768
24+d * 16+d =768
384 + 40d+d^2 = 768
d^2 + 40d-384 =0
(d+48) (d-8) = 0
d=-48, d=8 can't use a negative number so d = 8
check:
24+8=32, 16+8=24, 32x 24 = 768
so 8 feet is added to each side
how do you find the inverse of a 2x2 matrix
Answers
[tex] \frac{1}{determinate} \left[\begin{array}{ccc}d&-b\\-c&a\end{array}\right][/tex]
which is ad-bc
then use this to find the inverse
To find the inverse of a 2x2 matrix, calculate the determinant (ad-bc), then swap the diagonal elements, change the signs of the off-diagonal elements, and multiply each by the reciprocal of the determinant.
Explanation:To find the inverse of a 2x2 matrix, you must follow a specific procedure. Given a 2x2 matrix A:
\( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)
The inverse of matrix A, denoted as \( A^{-1} \), is calculated using the formula:
[tex]\( A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)[/tex]
Here, \( ad - bc \) is called the determinant of matrix A. For the inverse to exist, the determinant must not be zero. To calculate the inverse, you compute the determinant \( (ad - bc) \), then swap the elements of the diagonal positions (a and d), change the signs of the off-diagonal elements (b and c), and then multiply each element by \( \frac{1}{ad - bc} \).
For example, if you have a matrix:
[tex]\( A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \)[/tex]
The determinant is [tex]\( 4\cdot6 - 7\cdot2 = 24 - 14 = 10 \).[/tex]
The inverse of A is:
[tex]\( A^{-1} = \frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix} \)[/tex]
John is participating in a marathon that is 26.2 miles. His distance (d, in miles) depends on his time (t, in hours). Which is an appropriate range for this situation?
Answers
The appropriate range for John's distance in miles (d) during the marathon is A. [tex]$0 \leq d \leq 26.2$[/tex].
In a marathon, the distance (d) John covers depends on the time (t) he spends running. The distance is fixed at 26.2 miles, so we need to find the appropriate range for the time (t) he spends running.
Let's calculate John's average speed (v) during the marathon. We know that speed is given by:
[tex]\[ v = \frac{d}{t} \][/tex]
Where:
- v = average speed (miles per hour)
- d = distance covered (miles)
- t = time spent running (hours)
Given that John's distance is 26.2 miles, and the marathon covers this distance, we have:
[tex]\[ 26.2 = \frac{26.2}{t} \][/tex]
Solving for t:
[tex]\[ t = \frac{26.2}{26.2} = 1 \][/tex]
So, John takes 1 hour to cover the 26.2 miles.
Now, let's consider the maximum and minimum possible times for John to complete the marathon:
- Minimum time: John completes the marathon in the fastest time possible. Let's say this is 0. This implies he runs the marathon in 0 hours.
- Maximum time: John takes his time and completes the marathon at the slowest pace possible. Let's use the average time for a marathon, which is around 4.5 hours.
Thus, the appropriate range for the time (t) is:
[tex]\[ 0 \leq t \leq 4.5 \][/tex]
This corresponds to option C: [tex]$0 \leq t \leq 4.5$[/tex].
Complete Question:
John is participating in a marathon that is 26.2 miles. His distance (d, in miles) depends on his time (t, in hours) Which is an appropriate range for this situation?
A. [tex]$0 \leq d \leq 26.2$[/tex]
B. [tex]$0 \leq d \leq 4.5$[/tex]
c. [tex]$0 \leq t \leq 4.5$[/tex]
D. [tex]$0 \leq t \leq 26.2$[/tex]
Use the the factor theorem to determine wether the first polynomial is a factor of the second. X-3; 2x^2-4x+30
Answers
[tex]P(x)=2 x^{2} -4x+30[/tex]
If (x-3) is a factor of P(x), then
[tex]P(x)=2 x^{2} -4x+30=(x-3)*Q(x)[/tex], for some polynomial Q of 1st degree,
Then according to the factor theorem P(3)=0, because P(3)=(3-3)Q(x)=0*Q(3)=0.
Check
[tex]P(3)=2 (3)^{2} -4(3)+30=18-12+30=36[/tex]≠0
we see that P(3) is not 0, so (x-3) is not a factor of P(x).
Answer: no
Which theorem could Chelsea use to show the measure of angle KPR is equal to the measure of angle QRL?
Answers
can you help me????????
Answers
now, if AC is that much, recall, BD is the midsegment, thus AB = BC, so AB is just one of the equal halves of AC, so, AB is AC/2.
Consider the relation y = 4|x + 2| + 7. What are the coordinates of the vertex?
(7, −2)
(2, 7)
(4, −2)
(−2, 7)
Answers
x + 2 = 0
Then, we solve for the value of x.
x + 2 - 2 = 0 - 2
Hence, the value of x from the equation above is equal to -2.
Then, substitute the value of x to the equation and drop the absolute value symbol.
y = 4(x + 2) + 7
Substituting,
y = 4(-2 + 2) + 7
y = 4(0) + 7
y = 7
Thus, the vertex of the absolute value equation is equal to (-2,7). The answer is the last choice.
Conditional probabilities are based on some event occurring given that something else has already occurred?
Answers
The answer is true. A conditional probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A in the condition B", is usually written as P (A|B). The conditional probability of A given B is well-defined as the quotient of the probability of the joint of events A and B, and the probability of B.
What is the solution of sqrt 2x + 4 = 16 ? x = 6 x = 72 x = 126 no solution
Answers
Answer: Third option is correct.
Step-by-step explanation:
Since we have given that
[tex]\sqrt{2x+4}=16[/tex]
We need to find the value of 'x'.
First we squaring the both sides:
[tex](\sqrt{2x+4})^2=16^2\\\\2x+4=256\\\\2x=256-4\\\\2x=252\\\\x=\dfrac{252}{2}\\\\x=126[/tex]
Hence, the value of x is 126.
Therefore, Third option is correct.
Answer:
C on Edge
Step-by-step explanation:
Received a 100% on the quiz.
The sum of twice a number and a larger number is 145. The difference between the numbers is 55. Let x represent the smaller number and y represent the larger number. Which equations represent the situation? Check all that apply.
A. x-y=55
B. 2(x+y)=145
C. 2x+y=145
D. y-x=55
E. y=x+55
Answers
A does not, because x is actually 55 smaller than y, which is why both D and E work.
B does not because it's doubling the sum of x = y, but the description says to double x and then add y (which is why C works).
y>x
sum (addition) of twice a number (2x) and larger number (y) is (=) 145
2x+y=145
difference between them is 55 (y is bigger so it would be x is subtracted from y)
y-x=55
so the equations are
2x+y=145 and
y-x=55
we could add x to both sides in the 2nd equation to obtain
y=x+55, but that doesn't really represent that the difference is 55, though they are the same euation
so I would say C and D only
Find the surface area of a sphere with a volume of 36π in3.
SHOW WORK
will give medals and mark brainliest
its 113.10 inches squared
Answers
[tex]V=\frac{4}{3}\pi r^{3}[/tex]
[tex]r=\sqrt[3]{\frac{3V}{4\pi}}[/tex]
[tex]r=\sqrt[3]{\frac{3(36\pi)}{4\pi}}[/tex]
[tex]r=\sqrt[3]{27}[/tex]
[tex]r=3[/tex]
Then use the radius to get the surface area:
[tex]A_{surf}=4 \pi r^{2}[/tex]
[tex]A_{surf}=4 \pi 3^{2}[/tex]
[tex]A_{surf}=4 \pi 9[/tex]
[tex]A_{surf}=36\pi\approx113.09[/tex]
To find the surface area of a sphere given the volume, first solve for the radius using the volume formula (4/3πr³). In this case, the radius is 3. Then, use the surface area formula (4πr²), which gives the surface area as 36π square inches, or an approximate value of 113.10 square inches.
Explanation:The volume of a sphere is given by the formula V = 4/3 * π * r³. First, we need to find the radius of the sphere. Set the volume of the sphere (36π in³) equal to the volume formula and solve for r:
36π = 4/3πr³
From this, we find that r = 3. Now, we use the radius to find the surface area with the formula A = 4πr²:
A = 4π * 3² = 36π in²
So, the surface area of a sphere with a volume of 36π in³ is 36π square inches or approximately 113.10 square inches.
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Michelle found a new violin on sale at 30% off. how much would she pay the cashier if it originally sells for $250 in a city that had no sales tax
Answers
100%-30% = 70%
70%=0.70
250*0.70 = 175
she would pay $175
The sum of two numbers, x and y, is 12. The difference of x and two times y is 6. What are the values of x and y? x = 8, y = 4 x = 10, y = 2 x = 18, y = -6 x = 20, y = -8
Answers
x-2y=6
hmm
multiply 2nd equation by -1 and add to first
-x+2y=-6
x+y=12 +
0x+3y=6
3y=6
y=2
sub back
x+y=12
x+2=12
x=10
x=10, y=2
In the diagram, ∠ABC = 90°. What is the radius of the circle?
A. 5.7 in
B. 16.5 in
C. 24.6 in
D. 12.3 in
Answers
D. 12.3 in
Explanation and solution;We are give that ∠ABC = 90°; therefore a line from point A to point C is the diameter, this is because a diameter subtends an right angle to the circumference of the circle. Therefore; triangle ABC is a right-angled triangle, thus AC is the hypotenuse.Using the Pythagoras theorem;
AC² = AB² + BC²
= 22.1² + 10.9²
= 607.22
AC = √ 607.22
= 24.64
But, since AC is the diameter and the radius is half of the diameter, then
Radius = 24.64/2
= 12.32
≈ 12.3 (to 1 decimal place)
Which answer is correct
Answers
diagonal is square root of A^2+b^2
so C is the correct answer
You roll two standard number cubes. What is the probability that the sum is odd, given than one of the number cubes shows a 1? Show your work.
Answers
This probability is different because they give us a restriction with the "given that one f the cubes shows a one". The denominator becomes the number total number of rolls with a 1, and the numerator is the sum is odd.
P (odd sum is, given 1 number is a 1) = 6/11
The local theater has three types of seats for broadway plays: main floor, balcony, and mezzanine. main floor tickets are $59, balcony tickets are $50, and mezzanine tickets are $40. one particular night, sales totaled $73,785. there were 435 more main floor tickets sold than balcony and mezzanine tickets combined. the number of balcony tickets sold is 78 more than 33 times the number of mezzanine tickets sold. how many of each type of ticket were sold?
Answers
Final answer:
10 mezzanine tickets, 408 balcony tickets, and 853 main floor tickets were sold.
Explanation:
Let's solve this problem step-by-step to find out how many of each type of ticket were sold:
Let's assume that the number of mezzanine tickets sold is x. Therefore, the number of balcony tickets sold is 33x + 78 (since it is 78 more than 33 times the number of mezzanine tickets sold).
The number of main floor tickets sold is 435 + (33x + 78) + x = 435 + 34x + 78 = 34x + 513 (since there were 435 more main floor tickets sold than balcony and mezzanine tickets combined).
The total sales amount is $73,785.
Now, we can set up an equation to solve for x:
$40x + $50(33x + 78) + $59(34x + 513) = $73,785
Simplifying the equation:
40x + 1650x + 3900 + 59(34x + 513) = 73785
40x + 1650x + 3900 + 2006x + 30567 = 73785
3696x + 34467 = 73785
3696x = 39318
x = 39318/3696
x = 10.65
Since we can't have a fraction of a ticket, we can round down to the nearest whole number. So, x = 10.
Therefore, 10 mezzanine tickets were sold, 33x + 78 = 408 balcony tickets were sold, and 34x + 513 = 853 main floor tickets were sold.
Students were surveyed about their preference between dogs and cats. The following two-way table displays data for the sample of students who responded to the survey.
Approximately what percent of students in the sample were male?
Round your answer to the nearest percent.
%
Preference Male Female TOTAL
Prefers dogs 36 20 56
Prefers cats 10 26 36
No preference 2 6 8
TOTAL 48 52 100
Answers
To find the percentage of students in the sample who were male, divide the total number of male students by the total number of students and multiply by 100.
Explanation:To find the percentage of students in the sample who were male, we need to look at the total number of male students and divide it by the total number of students in the sample. From the given two-way table, we can see that the total number of male students is 48. The total number of students in the sample is 100. To find the percentage, we can divide 48 by 100 and multiply by 100 to get:
Percentage of male students = (48/100) * 100 = 48%
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Answer:
36%
Step-by-step explanation: