Suppose you have 52 feet of fencing to enclose a rectangular dog pen. The function A = 26x – x2, where x = width, gives you the area of the dog pen in square feet. What width gives you the maximum area? What is the maximum area? Round to the nearest tenth as necessary. width = 26 ft; area = 364 ft2 width = 26 ft; area = 169 ft2 width = 13 ft; area = 507 ft2 width = 13 ft; area = 169 ft2
Answers
Answer:
width = 13 ft; area = 169 ft^2
Step-by-step explanation:
The equation for the area can be factored as ...
A = x(26 -x)
This has zeros at x=0 and x=26. The vertex of this downward-opening parabola will be halfway between those, at x=13. Then the area for a width of 13 feet is ...
A = 13(26 -13) = 13^2 = 169 . . . . ft^2
The maximum area is 169 ft^2, when the width is 13 ft.
_____
You will note that these dimensions make the pen be a square. It can be useful to remember that a square is the rectangle with the largest area for a given perimeter length.
The width that gives the maximum area for the given fencing is 13 feet, and the maximum area is 169 square feet (option B).
Explanation:
The formula given is a quadratic function where x is the width and A is the area. In a quadratic function, the maximum or minimum value occurs at the vertex. The x-coordinate of the vertex can be found using the formula -b/2a where a and b are the coefficients of the quadratic function. Here, a is -1 and b is 26, therefore the x-coordinate of the vertex is -26/-2*-1 = 13. This means that a width of 13 ft will give the maximum area. To find the maximum area, we substitute x = 13 into the function: A = 26*13 - 13^2 = 169 ft2. So, the width that will give the maximum area is 13 feet and the maximum area is 169 square feet.
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Related Questions
You watch a roulette wheel spin 8 consecutive times and the ball lands on a red slot each time. what is the probability that the ball will land on a red slot on the next spin?
Answers
Answer:
0.5
Step-by-step explanation:
A roulette wheel is half red and half black. Each spin is independent: past spins don't change the probability of future spins. So even if you got 8 reds in a row, the probability of the next spin being red is still 50/50.
When probability is not found by considering all possible outcomes, but by testing and experimentation, it is called _____ probability.
Answers
Answer:
Step-by-step explanation:
Observational probability
Arlo invested $4000 in an account that earns 5.5% interest, compounded annually. The formula for compound interest is A(t)=P(1+i)^t. How much did Arlo have in the account after 4 years?
Answers
[tex]\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$4000\\ r=rate\to 5.5\%\to \frac{5.5}{100}\dotfill &0.055\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &4 \end{cases} \\\\\\ A=4000\left(1+\frac{0.055}{1}\right)^{1\cdot 4}\implies A=4000(1.055)^4\implies A\approx 4955.2986[/tex]
Answer:
$ 4955.30 ( approx )
Step-by-step explanation:
The formula for compound interest is,
[tex] A(t)=P(1+i)^t[/tex]
Where, P is the principal amount,
i is the rate per period,
t is the number of periods,
Here, P = $ 4000,
i = 5.5% = 0.055
t = 4 years,
By substituting the values,
The amount in the account after 4 years would be,
[tex]A=4000(1+0.055)^4=4000(1.055)^4=4955.2986025\approx \$4955.30[/tex]
Find four solutions of the given function. Write the solutions as ordered pairs.
4x – y = 4
Answers
The four solutions to the function 4x - y = 4 are determined by selecting different x values and calculating the corresponding y values, resulting in the ordered pairs (0, -4), (1, 0), (2, 4), and (3, 8).
Explanation:To find four solutions to the function 4x – y = 4, we need to choose four different values for x and compute the corresponding y values.
For example, let's choose x = 0, x = 1, x = 2, and x = 3. Substituting these values into the equation:
For x = 0: 4 * 0 - y = 4, so y = -4 For x = 1: 4 * 1 - y = 4, so y = 0 For x = 2: 4 * 2 - y = 4, so y = 4 For x = 3: 4 * 3 - y = 4, so y = 8
So the four solutions (order pairs) are (0, -4), (1, 0), (2, 4), and (3, 8).
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To find solutions for the given equation 4x - y = 4, pick x values and plug them into the equation to solve for y, resulting in ordered pairs like (0, -4) and (1, 0).
Given equation: 4x - y = 4
Choose x values and plug them into the equation to solve for y.
Write the solutions as ordered pairs (x, y).
Four solutions are (0, -4), (1, 0), (2, 4), and (3, 8).
The probability that a random variable is greater than or equal to z standard deviations from the mean in a standard normal distribution is p%. What can be said with certainty about the probability that the random variable is less than or equal to –z standard deviations from the mean?
Answers
Answer:
The right answer is B: The probability is equal to p%.
Step-by-step explanation:
The perimeter of a rectangular garden is 64 centimeters. The length of the garden is three times the width. Find the length
and the width
Answers
Answer:
width 8 centimeters
length 24 centimeters
Step-by-step explanation:
L=3W : equation a .
W=64/xW : equation b .
x=2L+2W .
=2(3W)+2W .
x=8W .
Answer:
Length = 24 cm
and, Width = 8 cm
Step-by-step explanation:
It is given that,
Length = 3 × (width)
⇒ L = 3W → (1)
Also, since perimeter = 64 cm
And perimeter of rectangular garden = 2(L + W) = 64
From equation (1)
2(3W + W) = 64
2 × 4W = 64
⇒ 8W = 64
⇒ W = 8 cm
Putting value in equation (1)
L = 3 × 8 = 24 cm
thus, Length = 24 cm
and, Width = 8 cm
Solve log (7x + 7) = 1. Round to the nearest thousandth if necessary. 0.429 2.333 1.143 0.875
Answers
Step-by-step explanation:
log(7·x + 7) = 1
7·x + 7 = 10^1
7·x = 10 - 7
x = 3/7 = 0.4285714285
Answer:
0.429
Step-by-step explanation:
Research scientists need a certain type of bacteria to conduct an experiment. There are 2,500 bacteria in a certain culture. The culture grows at a rate of 25% daily. The scientists needs at least 10,000 bacteria to conduct an experiment. What is the least number of days they need to wait for the bacteria culture to reach a quantity of 10,000? (Hint: use a guess-and-check method to determine the lowest number of days that satisfy the requirement.)
Answers
Answer:
7 days
Step-by-step explanation:
At 25% per day, it will take approximately 3 days to double the population, so approximately 6 days for the population to quadruple. Checking that number, we find it is not quite enough for the experiment, so another day is required.
"Guess and check" as a method of solution works especially well if you have an automated checker to evaluate your guess. A graphing calculator or spreadsheet can work well for this.
_____
We guess 3 days as the doubling time using the "rule of 72" that says the product of percentage and doubling time is about 72. That is, 72/25 ≈ 3. (This is only a very rough approximation of doubling time, best for rates near 8%.)
4. Divide –50m3n5 by –5m2n2.
A. 10mn3
B. –45m5n7
C. –10mn3
D. 45m5n7
Answers
Answer:
A. 10mn³
Step-by-step explanation:
To divide numbers in power form but to the same base, we simply subtract the powers and then divide the coefficients of the the bases.
-50m³n⁵ /-5m²n²
=10m⁽³⁻²⁾ n⁽⁵⁻²⁾
=10mn³
A negative number divided by a negative number = a positive number.
Is it true that since sin^2x + cos^2x = 1, then sin(x) + cos(x) = 1? Explain your answer.
Answers
Answer:
No it is not true because plugging in something like x = pi/4 radians (equivalent to 45 degrees) leads to the left side not being equal to 1. The left side will simplify to sqrt(2). So the equation is not true when x = pi/4 radians. There are infinitely other counter examples to use
Which equation is equivalent to
log5x^3 - logx^2 = 2?
Answers
Answer:
the second option and x=20
Answer:
the second one and then x=20
Step-by-step explanation:
What he said. It's correct.
Since we have computer algebra systems that can solve polynomial division problems for us, why is it necessary to learn how to do these things by hand?
Answers
Answer:
Probably because learning to do polynomials by hand boosts understanding of the topic and improves general algebraic solving ability,both of which are required in further topics like advanced calculus and algebra
Learning polynomial division by hand is crucial for understanding concepts, enhancing problem-solving skills, and fostering critical thinking in mathematics.
Explanation:The importance of learning polynomial division by hand despite the availability of computer algebra systems lies in developing a deep understanding of the underlying concepts, fostering problem-solving skills, and enabling visualization of geometric interpretations.
By manually carrying out mathematical operations, students can better grasp the logic behind the processes, which aids in building a strong foundation for more complex mathematical tasks.
Furthermore, mastering hand calculations equips students with the ability to perform quick mental math, apply critical thinking skills, and comprehend the practical applications of the mathematical concepts.
Find the slope of the line on the graph. Write your answer as a fraction or a whole number, not a mixed number or decimal.
Answers
The rise from one point to another is 2, and the run is 3 and since slope is rise/run, the slope is 2/3
Answer: The slope of the line is: [tex]\frac{2}{3}[/tex]
Step-by-step explanation:
The slope can be calculated with this formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
You need to choose two points of the line. You can pick the point (6,2) and the point (-6,-6)
You can identify that:
[tex]y_2=-6\\y_1=2\\x_2=-6\\x_1=6[/tex]
Now, you must substitute these values into the formula:
[tex]m=\frac{-6-2}{-6-6}[/tex]
With this procedure you get that the the slope of the line on the graph is:
[tex]m=\frac{-8}{-12}[/tex]
[tex]m=\frac{2}{3}[/tex]
Solve using elimination PLEASE I NEED HELP!!! IT WOULD REALLY MEAN A LOT IF ANYONE CAN ANSWER THIS! 16 POINTS!!!
2x+4y=1
x-4y=5
Answers
Answer:
(x, y) = (2, -3/4)
Step-by-step explanation:
The point of the "elimination" technique is to combine the equations in a way that eliminates one of the variables. Sometimes this involves multiplying one or both of the equations by constants before you add those results together. In any event, the first step is to look at the coefficients of the variable terms to see if there is a simple combination of them that will result in zero.
The y terms have coefficients that are opposites of each other (4, -4), so you can simply add the two equations to eliminate y as a variable.
(2x +4y) +(x -4y) = (1) +(5)
3x = 6 . . . . . simplify
x = 2 . . . . . . divide by 3
Now, you find y by substituting this value into one of the equations. I would choose the equation with the positive y-coefficient:
2(2) +4y = 1
4y = -3 . . . . . . subtract 4
y = -3/4
Then the solution is ...
(x, y) = (2, -3/4)
_____
A graphing calculator confirms this solution.
PLEASE HELP ME!!!
Find p(–5) and p(3) for the function p(x) = 2x5 – 9x4 – 2x2 + 12x – 2. Question 8 options: –11,915; –251 –4,487; –551 –11,985; –225 –11,987; –227
Answers
Answer:
p(-5)=-11,987
p(3)=-227
Step-by-step explanation:
To find the value of the function p(-5) just substitute x=-5 into the function expression instead of x:
[tex]p(-5)=2\cdot (-5)^5-9\cdot (-5)^4-2\cdot (-5)^2+12\cdot (-5)-2=\\ \\=-6,250-5,625-50-60-2=-11,987[/tex]
To find the value of the function p(3) just substitute x=3 into the function expression instead of x:
[tex]p(3)=2\cdot 3^5-9\cdot 3^4-2\cdot 3^2+12\cdot 3-2=\\ \\=486-729-18+36-2=-227[/tex]
Answer with Step-by-step explanation:
We are given a function:
[tex]p(x) = 2x^5-9x^4-2x^2+12x-2[/tex]
We have to find the value of p(-5) and p(3)
[tex]p(-5) = 2\times (-5)^5-9\times (-5)^4-2\times (-5)^2+12\times (-5)-2\\\\=-6250-5625-50-60-2\\\\=-11987[/tex]
[tex]p(3) = 2\times 3^5-9\times 3^4-2\times 3^2+12\times 3-2\\\\=486-729-18+36-2\\\\=-227[/tex]
Hence, p(-5)= -11987
and p(3)= -227
Adelia drove from her house to Townsville one evening. Her distance was 180 miles. She left home at 6:27 and was at townsville at 6:42 what was her speed
Answers
Answer: 720 mph or 12 miles a minute
Step-by-step explanation:
Answer:
60
Step-by-step explanation:
(I actually have the problem in front of me, and I think you switched the minute and hour hands on the clock)
Adel
left at 5:30 and arrived at 8:30. That’s 3 hours, which is 180 minutes. 180/180 = 1 mile per minute
60 minutes in an hour
60*1 = 60 mph
The rabbit population in a small zoo starts with 150 rabbits. The rabbit population is decreasing at a rate of 15% each week. Which of the following represents the total number of fish, F(x), after x number of weeks. (4.2)
a. F(x) = 150( .15)^x
b. F(x) = 150 (1.15)^x
c. F(x) = .85 (150)^x
d. F(x) = 150 (.85)^x
e. None of the above.
Answers
Answer:
e. none of the above
Step-by-step explanation:
I had this question as well and if you look closely, the question describes the function in terms of rabbits, when the actual question describes fish. I believe that because the question is asking for fish when only information on rabbits has been given, the only logical answer would be e. none of the above...
Final answer:
The correct formula representing the total number of rabbits after x weeks, with a starting population of 150 and a decrease of 15% per week, is F(x) = 150 (0.85)ˣ. The correct option is d.
Explanation:
The rabbit population in a small zoo starts with 150 rabbits and decreases at a rate of 15% each week. To represent the total number of rabbits, F(x), after x number of weeks, we need to find the correct formula. A decrease of 15% per week is the same as the population being at 85% of the previous week's population (100% - 15% = 85%). So, the correct formula would be the initial population multiplied by the weekly percentage to the power of the number of weeks. Therefore, the correct formula is F(x) = 150 (0.85)ˣ, which corresponds to option d.
Find all solutions of the equation. leave answers in trigonometric form x^3-8=0 2cis
Answers
[tex]x^3-8=0[/tex]
[tex]x^3=8=8\mathrm{cis}\,0^\circ[/tex]
By DeMoivre's theorem, we have
[tex]x=8^{1/3}\mathrm{cis}\left(\dfrac{0^\circ+360^\circ k}3\right)[/tex]
with [tex]k=0,1,2[/tex]. Then we have three solutions,
[tex]x=2\mathrm{cis}\,0^\circ=2[/tex]
[tex]x=2\mathrm{cis}\,120^\circ[/tex]
[tex]x=2\mathrm{cis}\,240^\circ[/tex]
Given the parallelogram ABCD, solve for X when angle A equals (30+ 5x)° and angle D equals (15+10x)° degrees. Also, sides AB and CD opposite each other.
Answers
Answer:
x = 9
Step-by-step explanation:
Adjacent angles A and D are supplementary:
(30 +5x)° +(15 +10x)° = 180°
15x +45 = 180 . . . . . collect terms, divide by °
15x = 135 . . . . . . . . . subtract 45
135/15 = x = 9 . . . . . divide by the coefficient of x
The value of x is 9.
. Solve the following system of equations: 1. y - 3x + 15 = 10
2. 6 - 4x = y
Answers
The first equation can be further simplified so first do that by subtracting 15 to both sides
y - 3x + (15 - 15) = 10 - 15
y - 3x = -5
We know from the second equation that y is 6 - 4x so in the equation y - 3x = -5 plug in 6 - 4x for y
(6 - 4x) - 3x = -5
6 - 4x - 3x = -5
Combine like terms
6 - 7x = -5
(6 - 6) - 7x = -5 - 6
-7x = -11
Isolate x
-7x/-7 = -11/-7
x = 11/7
To solve for y plug in 11/7 in for x in the equation 6 - 4x = y
6 - 4(11/7) =y
6 - 44/7 = y
y = -2/7
so...
(11/7 , -2/7)
Hope this helped!
~Just a girl in love with Shawn Mendes
1.) Solving for X problem 1#
y-3x+15=10
-3x+15-15=10-15-y
-3x=-y-5
-3x/-3=-y-5/3
x=(-y-5)/3
2.) Solving for Y problem 1#
y-3x+15=10
y-3x+3x+15-15=10-15+3x
y=3x-5
1.) Solving for X problem 2#
6-4x=y
-4x+6-6=y-6
-4x/4=y-6/4
x=(y-6)/4
2.) Solving for Y problem 2#
6-4x=y
y=-4x+6
suppose f(x)=x^2-2 find the graph of f(1/2x)
Answers
Answer:
graph g(x)=1/4 x^2 - 2
Step-by-step explanation:
You are to replace x with (1/2x) in the expression x^2-2
So you have (1/2x)^2-2
1/4 x^2-2
Graph some points for g(x)=1/4 x^2-2
The vertex is (0,-2) and the parabola is open up.
I would graph 2 more points besides the vertex
x | g(x) ordered pairs to graph
----------- (-1,-1.75) and (0,-2) and (1,-1.75)
-1 -1.75
0 -2
1 -1.75
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I need help
You find 20 books that you checked out from the library as you are cleaning your room. You need to take 4 books back to the library, and you want to set them aside in a pile on your desk. Â
How many ways can 4Â of the 20Â books be arranged in a pile?
Answers
Answer:
the answer is 20P4 = 4845
Step-by-step explanation:
Answer:
There are 20 ways to pick the first book, 19 ways for the second
Step-by-step explanation:
If order of the 4 books does not matter then
₂₀C₄ = 20! / (20-4)!4! = 20!/(16!*4!) = 4845 ways
Help! the file is attached
Answers
Answer:
924/5 ft/s369.6 ftStep-by-step explanation:
126 mi/h = (126 mi)/(1 h)·(5280 ft)/(1 mi)·(1 h)/(3600 s) = (126·5280/3600) ft/s
126 mi/h = 184.8 ft/s = 924/5 ft/s
__
In 2 seconds, the parachutist falls ...
(184.8 ft/s)·(2 s) = 369.6 ft
A missile is launched from the ground. Its height,h(x) can be represented by a qyadratic function in terms of time, x, in seconds after 1 second the missile is 103 feet in the air; after 2 seconds it is 192 feet in the air. Find the height in feet of the missile after 5 seconds in the air
Answers
Answer:
375 ft
Step-by-step explanation:
The increase in height in the 1st second is 103 ft. In the 2nd second, it is 192-103 = 89 ft, a decrease of 14 ft. In the next three seconds, the increases in height can be expected to be ...
89 -14 = 75 ft
75 -14 = 61 ft
61 -14 = 47 ft
for a total increase in height over those 3 seconds of ...
75 + 61 + 47 = 183 ft
Then the height after 5 seconds in the air is ...
192 ft + 183 ft = 375 ft
_____
You can model the height function with the quadratic equation ...
h(t) = at^2 +bt
We need to find the values of "a" and "b", which we can do by substituting the given data point values. The given data is ...
h(1) = a + b = 103
h(2) = 4a + 2b = 192
Subtracting twice the first equation from the second, we get
(4a +2b) -2(a +b) = (192) -2(103)
2a = -14 . . . . . simplify
a = -7 . . . . . . . divide by 2
b = 103 -a = 110 . . . . find b using the first equation
Then the quadratic model of height is ...
h(t) = -7t^2 +110t
and the height at 5 seconds is ...
h(5) = -7·25 +110·5 = 375 . . . feet
Need help with a math question
Answers
Answer:
There are 10 possible ways the student can schedule the three classes
Step-by-step explanation:
The college student is choosing 3 classes to take during the first, second, and third semester from 5 electives.
Therefore, the student is choosing 3 items from a total of 5 and the order is not of key concern here. This is thus a combination problem;
5C3 = 10
read as 5 choose 3
The above mathematical operation can be performed using any modern calculator.
Use the figure to answer the questions
a. Explain why the triangles are similar
Geometry - Shaping Sheet
Triangle ______ and Triangle ______ are similar by the __________ postulate. This postulate works because __ (explain your reasoning here) ________.
*Use this format to appropriately write your answer below this box.
b. Find the value of x
Answers
Answer:
A. EAB and EDC by AAA because congruent corresponding angles lead to similar triangles regardless of side length. This postulate works because the parallel lines cause angle a and angle d to be congruent as well as angle b being congruent to angle c, and vertical angle e is congruent.
Step-by-step explanation:
How many combinations of 7 candidates can fill 3 vacancies on a city council?
Answers
Answer:
35 combinations
Step-by-step explanation:
You have to use the formula for finding combinations in probability. It looks like this:
₇C₃ = [tex]\frac{7!}{3!(7-3)!}[/tex]
so that gives you
₇C₃ = [tex]\frac{7*6*5*4!}{3*2*1*4!}[/tex]
The 4 factorials cancel each other out, and when you do the math, you get 210/6. That divides evenly into 35. So there are 35 combinations.
Choose the correct reason for each algebraic statement. A. Subtraction Property of Equality B. Combine like terms C. Distributive Property D. Division Property of Equality 3.__ 4.__ 5.__ 6.__
Answers
Answer:
b
Step-by-step explanation:
hope this helps
Find the smallest positive $n$ such that \begin{align*} n &\equiv 3 \pmod{4}, \\ n &\equiv 2 \pmod{5}, \\ n &\equiv 6 \pmod{7}. \end{align*}
Answers
4, 5, and 7 are mutually coprime, so you can use the Chinese remainder theorem right away.
We construct a number [tex]x[/tex] such that taking it mod 4, 5, and 7 leaves the desired remainders:
[tex]x=3\cdot5\cdot7+4\cdot2\cdot7+4\cdot5\cdot6[/tex]
Taken mod 4, the last two terms vanish and we have[tex]x\equiv3\cdot5\cdot7\equiv105\equiv1\pmod4[/tex]
so we multiply the first term by 3.
Taken mod 5, the first and last terms vanish and we have[tex]x\equiv4\cdot2\cdot7\equiv51\equiv1\pmod5[/tex]
so we multiply the second term by 2.
Taken mod 7, the first two terms vanish and we have[tex]x\equiv4\cdot5\cdot6\equiv120\equiv1\pmod7[/tex]
so we multiply the last term by 7.
Now,
[tex]x=3^2\cdot5\cdot7+4\cdot2^2\cdot7+4\cdot5\cdot6^2=1147[/tex]
By the CRT, the system of congruences has a general solution
[tex]n\equiv1147\pmod{4\cdot5\cdot7}\implies\boxed{n\equiv27\pmod{140}}[/tex]
or all integers [tex]27+140k[/tex], [tex]k\in\mathbb Z[/tex], the least (and positive) of which is 27.
On a number line the directed line segment from Q to S has endpoints Q at -14 and S at 2. Point R partitions the directed line segment from Q to S in a 3:5 ratio.
Which expression correctly used the formula, to find the location of point R?
Answers
Answer:
Step-by-step explanation:
The distance from Q to S is 2 - (-14), or 16.
We start at point Q. Note how 3 and 5 add up to 8, which allows us to write:
R = Q + (3/8)(16), or R = -14 + 6, or R = -8.
From R to S it is (5/8)(16), or 10 units.
The directed line segment is partitioned into segments of lengths 6 and 10, whose combined length is 16, as expected.
Answer:
14/5 or A
Step-by-step explanation:
I just took this assignment.