A wire 24inches long is to be cut into four pieces to form a rectangle whose shortest side has a length of x:
Determine the domain of the function and use a graphing utility to graph the function over that domain
Use the graph of the function to approximate the maximum area of the rectangle. Make a conjecture about the dimensions that yield a maximum area. ...?
Answers
0 < x<6, If x is 7 then width would be 5, but x must be shorter
Answer:
Area function : [tex]A(x)=12x-x^2[/tex]
Domain: (0,6)
The area of rectangle is maximum at x=6. The area of a rectangle is maximum if it is a square.
Step-by-step explanation:
It is given that the length of wire is 24 inches. It is to be cut into four pieces to form a rectangle.
Let x be the length of shortest side.
Perimeter of a rectangle is
Perimeter = 2( Shortest side + longest side).
[tex]24 = 2( x + \text{longest side})[/tex]
[tex]12 = x + \text{longest side}[/tex]
[tex]12 - x = \text{longest side}[/tex]
So, length of longest side is (12-x) inches.
Area of a rectangle is
[tex]A=length \times width[/tex]
Area function is
[tex]A(x)=x(12-x)[/tex]
The area of rectangle and dimensions of a rectangle can not be a negative.
[tex]A(x)>0[/tex]
[tex]x(12-x)>0[/tex]
It means,
[tex]x>0[/tex]
[tex]12-x>0\Rightarrow 12>x[/tex]
One side is less that the other side.
[tex]x<12-x[/tex]
[tex]2x<12[/tex]
[tex]x<6[/tex]
It means the domain of the function is (0,6).
The simplified form of the area function is
[tex]A(x)=12x-x^2[/tex]
Differentiate with respect to x.
[tex]A'(x)=12-2x[/tex]
[tex]A'(x)=0[/tex]
[tex]12-2x=0[/tex]
[tex]x=6[/tex]
Differentiate A'(x) with respect to x.
[tex]A''(x)=-2<0[/tex]
Therefore the area of rectangle is maximum at x=6.
[tex](12-x)=12-6=6[/tex]
It means the area of a rectangle is maximum if it is a square.
Related Questions
60% of what number is 13.5?
Answers
The required, 60 percent of 22.5 is equal to 13.5, as of the given condition.
What is percentage?The percentage is a way of expressing a fraction or a decimal number as a portion of 100. It is denoted by the symbol '%'. For example, the percentage of 0.75 can be calculated as 0.75 x 100% = 75%. In other words, we are expressing 0.75 as a fraction of 100.
Here,
To find out what number 60% of is equal to 13.5, we can use algebra. Let x be the number we are looking for. Then we can write:
60% of x = 13.5
We can convert 60% to a decimal by dividing by 100:
0.6x = 13.5
Finally, we can solve for x by dividing both sides of the equation by 0.6:
x = 13.5 ÷ 0.6
x = 22.5
Therefore, 60% of 22.5 is equal to 13.5.
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Simplify (4xy^-2)/(12x^(-1/3)y^-5) and Show work ...?
Answers
The fraction is: [tex] \frac{4xy^{-2} }{12 x^{-1/3} y^{-5} } [/tex]
Let's rewrite it: [tex]\frac{4xy^{-2} }{12 x^{-1/3} y^{-5} }=\frac{4 }{12 }*\frac{x }{ x^{-1/3} }*\frac{y^{-2} }{ y^{-5} }[/tex]
Since [tex] \frac{x^{a} }{x^{b}} =x^{a-b} [/tex],
then:
*** [tex]\frac{x}{ x^{-1/3} }= x^{1-(-1/3)} = x^{3/3+1/3} = x^{4/3} [/tex]
*** [tex]\frac{y^{-2} }{ y^{-5} }= y^{-2-(-5)} = y^{-2+5} = y^{3} [/tex]
______
[tex]\frac{4 }{12 }*\frac{x }{ x^{-1/3} }*\frac{y^{-2} }{ y^{-5} }= \frac{1*4}{3*4}* x^{4/3}*y^{3}= \frac{1}{3} * x^{4/3}*y^{3}= \frac{x^{4/3}y^{3}}{3} [/tex]
Use the Laplace transform to solve the given initial value problem. y' + 6y = e^4t ; y(0)=2 ...?
Answers
The Laplace transform method is applied to solve the differential equation y' + 6y = e^4t with the initial condition y(0)=2. After transforming, simplifying, and solving for Y(s), we use inverse Laplace transform to find the solution y(t) in the time domain.
Explanation:Laplace transform is a powerful tool in the field of mathematics used for solving differential equations. To solve the given initial value problem y' + 6y = e^4t ; y(0)=2, we can start by taking the Laplace transform of both sides of the equation.
The Laplace transform of y' is sY(s) - y(0) and the Laplace transform of y is Y(s). Therefore, the Laplace transform of y' + 6y gives sY(s) - y(0) + 6Y(s). Given that y(0)=2, this simplifies to sY(s) + 6Y(s) - 2.
On the right-hand side, the Laplace transform of e^4t is 1/(s-4). Thus, we have the equation sY(s) + 6Y(s) - 2 = 1/(s-4).
By solving for Y(s), we can find the inverse Laplace transform to get the solution y(t) in the time domain.
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4(x+3)=20
Help? Solve the equation
Answers
4x= 20-12
4x= 8
x= 8/4 = 2
multiply 4 time x and 4 time 3
4x+12=20
then you subtract 12 to 20 and is 8
4x=8
then you divide 8 by 4 and i
x=2
Is the ordered pair a solution to the system of linear equations?
Select Yes or No.
Ordered pair: (−2,2)(−2,2)
System of equations:
−7x+2y=0
6x+6y=0
Answers
6x + 6y = 0
ordered pair: (−2,2)
the way to figure out if that ordered pair is a solution is pretty simple, you just plug the ordered pair into both equations, and check if the outcome is true.
-7(-2) + 2(2) = 0
14 + 4 = 0
18 = 0 is NOT true, so this isn't a solution. still, check your second equation as well.
6x + 6y = 0
6(-2) + 6(2) = 0
-12 + 12 = 0
for this equation, it is a solution. however, because the solution must apply to the system as a whole, and not just one of the equations, the ordered pair still is not a solution.
What is 20 + 10? (Nobody answer this)
Answers
Remove the zeros and add normally. Then add 1 zero next to the answer.
Another way:
Add 20+10.
Answer: 30
If f(x) = 8, what is f(x) when x = 10? 4 9 13 36
Answers
If f(x) = 8, then f(x) when x = 10 is also 8, because the function's value is constant for all x.
Explanation:If the function f(x) is defined as f(x) = 8 for all values of x, then regardless of the value we substitute for x, the function's output remains constant at 8. So when x = 10, the value of f(x) is still 8. This is an example of a horizontal line on a graph where the y-value does not change regardless of the x-value. In terms of probability, if we're dealing with a continuous probability distribution where f(x) is a constant, then to find P(0 < x < a), we would simply evaluate the integral of f(x) over the interval from 0 to a. However, given that in this case f(x) is not a probability density function and is a constant value, P(x) would not be applicable.
What is 1036.50 rounded the the nearest whole number?
Answers
Fernando purchased 1 pound of dried cranberries and 4 pounds of almonds to make a trail mix. Cranberries cost $4 per pound and almonds cost $5 per pound. What is the price per pound of the trail mix?
Answers
4 lb alm: 20
24 dollars for 5 lb
24/5 is 4.8
$4.80 per lb
Write the number 7/10 as a decimal and describe the process
Answers
We got the answer by doing
7 divide 10
I hope this helps you :)
1735/1000 is what in decimal form
Answers
We divide now 1735 by 1000 what we write down as 1735/1000 and we get 1.735 1735/1000 as a decimal equals 1.735
Suppose that alpha and beta are int variables. The statement alpha = beta--; is equivalent to the statement(s) ____.
a. alpha = 1 - beta;
b. alpha = beta - 1;
c. beta = beta - 1;
alpha = beta;
d. alpha = beta;
beta = beta - 1
Answers
What is the average rate of change of the function f(x)=2(3)^x from x = 2 to x = 4?
Answers
The average rate of change of the given function will be 72.
What is the average rate of change of the function?It quantifies how much the function changed per unit on average throughout that time interval. The slope of the straight line connecting the interval's ends on the function's graph is used to calculate it.
The given function is f(x)=2(3)ˣ and the interval is x ∈ [2,4].
The formula to calculate the average rate of change is (f(b)-f(a))/(b-a).
Here, a = 2 and b = 4
So,f (2) = 2.3² = 18
Now,f (4) = 2.3⁴ = 162
The average rate of change = (162 - 18)/(4 - 2)
The average rate of change = 144/2
The average rate of change = 72
Therefore, the average rate of change of the given function will be 72.
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The average rate of change of the function f(x) = 2(3)ˣ from x = 2 to x = 4 is 72. This means the secant line that intersects the graph of the function at x = 2 and x = 4 has a slope of 72.
Step 1: Formula for Average Rate of Change
The average rate of change of a function f(x) over the interval [a, b] is calculated using the following formula:
Average rate of change = (f(b) - f(a)) / (b - a)
where:
f(b) is the function's value at the upper bound (x = 4 in this case).f(a) is the function's value at the lower bound (x = 2 in this case).b is the upper bound of the interval (x = 4).a is the lower bound of the interval (x = 2).Step 2: Find f(b) and f(a)
We are given the function f(x) = 2(3)ˣ . Let's find the function's values at x = 4 (upper bound) and x = 2 (lower bound):
f(4) = 2(3)⁴ = 2 × 81 = 162f(2) = 2(3)⁴ = 2 × 9 = 18Step 3: Calculate the Average Rate of Change
Now that we have f(b) and f(a), we can plug them into the formula along with the interval's bounds:
Average rate of change = (f(4) - f(2)) / (4 - 2)Average rate of change = (162 - 18) / (2)Average rate of change = 144 / 2Average rate of change = 72 Eric bought 300 t-shirts, which were sold in packs of 12. How many packs did Eric buy?
A) 20
B) 25
C) 28
D) 30
Answers
300 divided by 12 = 25.
Answer is B, 25, because 300 Divided 12 is 25 & 25x12=300.
What is the difference? -2/15 -(-9/15)
a.-11/15
b. -7/15
c. 7/15
d. 11/15
Answers
the answer is C
a cinder block is sitting on a platform 20 high. lt weighs 79n. The block has_____ energy. calculate it.
Answers
Calculations:
Ep = mgh = mg = Fg
Ep = (Fg)(h)
Ep = (79N)(20 m)
Ep = 1580 J.
I believe the block's potential energy with respect to gravity is 1580 J.
A population of 240 birds increases at a rate of 16% annually. Jemel writes an exponential function of the form f(x) = abx to represent the number of birds after x years. Which values should she use for a and b?
Answers
f ( x ) = a * b^x,
where x is the number of the years.
a = 240 ( number of the birds at start )
16 % = 0.16
b = 1 + 0.16 = 1.16
f ( x ) = 240 * 1.16^x
Answer:
a = 240, b = 1.16
Answer:
Hence the values of a and b are given by:
a=240
and b=1.16.
Step-by-step explanation:
It is given that:
A population of 240 birds increases at a rate of 16% annually.
i.e. the initial population of birds is 240.
Also they are increasing at a rate of 16% =0.16.
Hence, it clearly implies that the growth is exponential and the function that represents the population of the birds after x years is given by:
[tex]f(x)=240\times (1+0.16)^x\\\\\\f(x)=240\times (1.16)^x[/tex]
Hence, on comparing our function with the exponential function:
[tex]f(x)=ab^x[/tex]
we have:
a=240
and b=1.16.
The height of the Empire State Building is 318.00 meters. If a stone is dropped from the top of the building, what is the stone's velocity just before it strikes the ground?
Answers
Answer: In order to find the stone’s velocity just before it strikes the ground we will use one of the free fall equations. V = √(2gh). We already have the values needed which are the gravity (9.8 m/s2) and our total height (318 meters). All you have to do is solve for V. You should get 78.95 m/s.
Three interior angles of a quadrilateral measure 55°, 117°, and 120°. What is the measure of the fourth interior angle?
68°
78°
88°
98°
Answers
it would be: 360 - (55+117+120)
= 360 - 292
= 68 degrees
So, option A is your answer.
Hope this helps!
The measure of the fourth interior angle of the given quadrilateral is: B. 68°
What is the Sum of a Quadrilateral?The sum of all the interior angles of a quadrilateral equals 360 degrees.
The given interior angles of the quadrilateral are: 55°, 117°, and 120°
The measure of the fourth interior angle = 360 - 55 - 117 - 120
The measure of the fourth interior angle = 68 degrees.
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What is the middle term of the product of (5a - 7)(2a - 1)?
-4a
-19a
-24a
Answers
= 10a² - 14a - 5a + 7
= 10a² - 19a + 7
Middle term = -19a
Jim earns $1,600 per month after taxes. He is working on his budget and has the first three categories finished.
Housing $576
Food $272
Transportation $320
Why will he have a problem with the rest of his budget?
He is budgeting too much for transportation.
He has used the highest recommended percentages to calculate the amounts for the three categories.
He is budgeting too much for housing.
He has used the lowest recommended percentages to calculate the amounts for the three categories.
Answers
When budgeting, it is useful to use percentages slightly higher than the minimum to compensate for uncertainties.
B.
He has used the highest recommended percentages to calculate the amounts for the three categories.
write an explicit formula for sequence
The Given Sequence is..
3,-6,12,-24..
Answers
(-6).(-2)=12
12.(-2)=-24
.
.
.
q=-2
---------------
a(n)=a(1).q^(n-1)
a(n)=3.(-2)^(n-1)
Which of the following are equivalent to the function y = 4cos x - 2? Check all that apply.
A. y = -4cos x + 2
B. y = 4sin(x +π/2 ) - 2
C. y = 4sin(x - π/2) - 2
D. y = 4cos(-x) - 2 ...?
Answers
Out of the given options, B (y = 4sin(x +ext{ extpi}/2 ) - 2) and D (y = 4cos(-x) - 2) are equivalent to the function y = 4cos x - 2, due to trigonometric identities and properties.
To determine which of the given options are equivalent to the function y = 4cos x - 2, we need to look at trigonometric identities and properties of cosine and sine functions.
Option A: y = -4cos x + 2 is not equivalent since it represents a reflection in the x-axis followed by a vertical shift, which alters the original function's graph completely.Option B: y = 4sin(x +rac{ext{extpi}}{2}) - 2 is equivalent because of the co-function identity sin(x + rac{ext{ extpi}}{2}) = cos x, which makes the functions identical.Option C: y = 4sin(x - rac{ext{extpi}}{2}) - 2 is not equivalent because sin(x - rac{ext{extpi}}{2}) is not equal to cos x.Option D: y = 4cos(-x) - 2 is equivalent due to the even property of the cosine function, which states that cos(-x) = cos x.
The surface area of two similar solids are 340yd^2 and 1,158yd^2. The volume of the larger solid is 1,712yd^2. What is the volume of the smaller solid? ...?
Answers
170 percent converted to a fraction equals: ...?
Answers
After finding the simplest form by dividing 100 the fraction equivalent to 170 percent we get 17/10.
A figure or ratio that may be stated as a fraction of 100 is a percentage. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the word percent means.
Here we have to convert:
170 percent into a fraction
Divide it by 100 to condert into fractions,
170/100
Divide both the numerator and denominator by 10
17/10
Since 17 and 10 are co-primes,
Therefore, it is in simplest form.
Hene,
The fraction 17/10 is equivalent to 170%.
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Explain why all linear angle pairs must be supplementary, but all supplementary angles do not have to be linear pairs?
Answers
Linear angle pairs are adjacent angles (they are together) and they form a straight angle, which is 180°. Therefore, they are supplementary.
But two angles may be separated and yet add 180°, then they are supplementary but are not linear angle pairs.
Final answer:
Linear angle pairs are always supplementary and formed by intersecting lines, while supplementary angles can be any two angles that add up to 180 degrees.
Explanation:
In mathematics, linear angle pairs are formed by two adjacent angles that add up to 180 degrees. This is known as supplementary angles. Linear angle pairs always occur when two lines intersect, creating opposite angles or a straight angle, such as in the case of a triangle.
On the other hand, supplementary angles do not have to be linear pairs. Supplementary angles are any two angles that add up to 180 degrees, regardless of their position or relation to each other. They can be adjacent angles, non-adjacent angles, or angles across parallel lines.
For example, two angles measuring 60 degrees and 120 degrees are supplementary angles but not a linear pair, since they are not adjacent to each other or formed by intersecting lines.
Let p: The integer has one digit.
Let q: The integer is less than 10.
Which represents "The integer has one digit if and only if the integer is less than 10”?
p ∨ q
p ∧ q
p → q
p ↔ q
Answers
Answer:
p ↔ q
Step-by-step explanation:
Given :
Let p: The integer has one digit.
Let q: The integer is less than 10.
To Find:Which represents "The integer has one digit if and only if the integer is less than 10”?
Solution:
1. p ∧ q denotes “p and q”
2.p ∨ q denotes “p or q" (or both)
3.p → q denotes “if p then q”
4. Biconditional p ↔ q denotes “p if and only if q"
Thus Biconditional p ↔ q represents "The integer has one digit if and only if the integer is less than 10”
Hence Option D is correct p ↔ q
Which of the following is not a combination?
The number of ways a President and Vice President can be chosen from a group of 10 people.
The number of 2 person subcommittees that can be chosen from a 10 person committee.
The number of 2 person teams that can be picked form 10 people.
Answers
When you're picking people for specific roles, the order in which people are picked matters. That is a permutations situation, not combinations.
For the others, they're situations that can be modeled with combinations since order does not matter. All that matters when you make a subcommittee or a duo-team is who's in it.
When there is the no of ways where the president and the vice president should be selected from the grouping of 10 people does not represent the combination.
The following information should be considered:
In the case when the people are picked out for particular role so here the order should be matter. It should be the situation of permutation, not the combination situation. Based on this, the last 2 options are considered to be the combinationTherefore we can conclude that when there is the no of ways where the president and the vice president should be selected from the grouping of 10 people does not represent the combination.
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how are the integers and rational numbers different? ...?
Answers
Rational Numbers: Numbers that can be written as a fraction (or ratio) of two integers.
wholesale price $3.50 per item and is sold at a retail value of $5.00 what is the gross percentage of the item?