The dimensions of the rectangular window that gives a maximum area are as follows;
Width = [tex]\dfrac{12}{2+\pi }[/tex] feet and length = 3 feet
What is the maximum value of a function?The maximum value of a function is given by the point at which the graph of the function has the highest value.
The parameters of the window are;
Length of the framing material available = 12 ft.
The shape of the window = A semicircle on top of a regular rectangular window
Required; The dimension of the rectangular part of the Norman window that allows the most light
Solution;
Diameter of the semicircular part of the window = The width of the rectangular part
Let x represent the width of the window, we have;
Perimeter of the semicircle = π·x/2
Side length of the rectangular window = (12 - π·x/2 - x)/2
Area of the rectangular part of the window, A = x × (12 - π·x/2 - x)/2
Area of the semicircle = π·x²÷4
Area of the window, is therefore;
[tex]A =\dfrac{ x \times (12 - \dfrac{\pi \cdot x}{2} - x)}{2} +\dfrac{ \pi \cdot x^2}{4}[/tex]
At the maximum area, of the rectangular part, we have;
[tex]A_r' =\dfrac{d}{dx}\left( A =\dfrac{ x \times (12 - \dfrac{\pi \cdot x}{2} - x)}{2} \right) = -\dfrac{x\cdot \pi +2\cdot x-12}{2} = 0[/tex]
x·π + 2·x - 12 = 0
The width, x = [tex]\dfrac{12}{2+\pi }[/tex] feet
The length of the rectangle is therefore;
[tex](12 - \pi \cdot \dfrac{6}{2+\pi } - \dfrac{12}{2+\pi })/2 = 6 - \pi \cdot \dfrac{3}{2+\pi } - \dfrac{6}{2+\pi } = 3[/tex]
The length of the rectangular window = 3 feet
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4 yd.
3 yd.
What is the length of the hypotenuse? If necessary, round to the nearest tenth.
yards
Answer:
5 yds
Step-by-step explanation:
i am assuming that 4 yd and 3 yd are the legs
so
to find the hypotenuse you square both legs add them and then take the square root of that
so
4^2 + 3^2 = c^2
16 + 9 = c^2
25 = c^2
c = 5
the final side is 5
You need 1.25 cups of sugar to make 1 dozen cookies. If you want to make 78 cookies, how much sugar is needed? Round your answer to 1 decimal places as needed To make 78 cookies, you will need cups of sugar.
The quantity of sugar required to make 78 cookies is 8.1 cups.
Given that, you need 1.25 cups of sugar to make 1 dozen cookies.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Here,
The quantity of sugar to make 1 cookie
= 1.25/12
= 0.10417
The quantity of sugar to make 78 cookies
= 78×0.10417
= 8.1 cups
Therefore, the quantity of sugar required to make 78 cookies is 8.1 cups.
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Which is the BEST estimate of the average rate of change for the function graphed, over the interval 1 ≤ x ≤ 3?
A) 2
B) 3
C) 4
D) 6
The BEST estimate of the average rate of change for the function graphed, over the interval 1 ≤ x ≤ 3 is 2 so option A is correct.
Given:
1 ≤ x ≤ 3
we know x is between 1 and 3 so the best estimate is 2 even though the answer could also be 3 but that will be skewed.
1 ≤ 2 ≤ 3 is the best estimate.
Therefore the BEST estimate of the average rate of change for the function graphed, over the interval 1 ≤ x ≤ 3 is 2 so option A is correct.
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Johnny recently hired a cleaning company to clean his home. The cleaning company charges
a service fee of $46 and $12 for each cleaning visit. Johnny has a monthly cleaning budget of
$130. How many cleaning visits can Johnny afford? answer?
Find the perimeter of the quadrilateral. If x=2, the perimeter is (blank) inches.
Answer:
the perimeter is 129 inches
Step-by-step explanation:
4x² + 8x + 3x² - 5x + 20 + 7x + 30 + 31 =
= 7x² + 10x + 81
x = 2
= 7(2²) + 10(2) + 81
= 7(4) + 20 + 81
= 28 + 101
= 129
Answer: 129 in.
The sum of the first k positive integers is 990. What is k?
Pls give explaintion
Answer:
k=44
Step-by-step explanation:
1 + 2 + 3 + 4......+ k = 990
Arithmetic sequence with d = 1 a1 = 1
1 + 1+ 1 + 1 +2 ...... 1+ (k-1) = 990
ak = a1 + (k-1) <====formula for sequence
Sum of a sequence formula = Sk = k ( a1+ak)/2
990 = k ( 1 + (1 + k-1) /2
1980 = k^2 + k
k^2 + k - 1980 = 0 Use quadratic formula a = 1 b = 1 c = - 1980
to find k = 44
I can determine if a relation is a
function.
1. Use vertical line test to determine if
the graph is a function. Why or why not?
71
(
7
2
Z
2
4
Answer:
1st Graph : Function
2nd Graph : Not a Function
Step-by-step explanation:
For the 1st Graph , any Vertical Line will not pass the Graph in more than one Point , So it represents a Function
For the 2nd Graph , many Vertical Line will pass the Graph in two Points , So it is not a Function
5. At PrintAll copying, a new copying machine can produce five more than twice the number of copies per hour of the old copy machine. If the new machine produces 205 copies per hour, how many copies can the old machine produce? Show the equation you used to determine the answers.
6. A recipe calls for 2 cups of milk for every 6 1/4 cups of flour. If you increase the flour to 43 3/4 cups, how many cups of milk will you use?
Step-by-step explanation:
5.
xnew = 205 = 2×xold + 5
200 = 2×xold
xold = 100
the old city machine can produce 100 copies per hour.
6.
2 cups of milk relate to 6 1/4 cups of flour.
such recipes represent a direct dependency of one variable on the other. one variable is always the other variable multiplied by a constant factor.
m = cups of milk
f = cups of flour
m = n×f
2 = n×(6 1/4) = n×(6×4 + 1)/4 = n×25/4
8 = 25n
n = 8/25
now, f = 43 3/4 = (43×4 + 3)/4 = 175/4
so,
m = 8/25 × 175/4 = (8×175)/(25×4) = 2×7 = 14
you will need 14 cups of milk.
The graph below shows the daily low temperatures for one week in New Orleans, Louisiana. The shape of the temperature graph can be modeled by the quadratic function f (x) = 3x² 18x + 56, where 2 is the number of days starting Sunday, and f (2) is the temperature in Fahrenheit. Degrees Fahrenheit 60 50 40 30 20 10 0 - Sun. Mon. Tues Wed. Thu. Fri. Sat. a. Use the given quadratic function to approximate the temperature on Thursday. How does this approximation agree with the graph? b. Use the function and the quadratic formula to find when the temperature is 35 degrees Fahrenhait. (Hint: Set f (x) = 35 and solve.) Round the answer to one decimal place and interpret the decimal in terms of weekday and military time.
a. The numeric value at Thursday -> x = 4 is of 32 degrees, which agrees with the graph.
b. A numeric value of 35 degrees Fahrenheit is obtained on Monday and on Thursday, also agreeing with the graph.
How to obtain the numeric value of a function or of an expression?To obtain the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
In this problem, the function is defined as follows:
f(x) = 3x² - 18x + 56.
Giving the temperature in degrees Fahrenheit in x days after Sunday.
The day that represents Thursday is given by:
x = 4.
Hence the temperature is the numeric value of the function at x = 4, replacing both instances of x on the function by 4, hence:
f(4) = 3(4)² - 18(4) + 56 = 32 ºF.
Which agrees with the graph, as the Thursday's bar is a value slightly above 30, which can be interpreted as 32.
The numeric value is of 35 when:
3x² - 18x + 56 = 35
The solutions to the quadratic equation are:
x = 1.6, x = 4.4.
Hence the days are:
Monday and Thursday.
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Kurt spen $96.39 over 4(1)/(4) months on movie streaming services. If he spent the same amount of money each month, how much money in dollars, did Kurt spend per month on movie streaming services?
Kurt spent a total amount of $22.68 a month on movie streaming services.
What is a proportion?A proportion is a fraction of a total amount, and equations can be built to find the desired measures in the problem using a rule of three, or operations such as multiplication and division.
These operations are used from the unit rates, as when we have an amount over some time, then the amount per unit of time is the total amount divided by the total time.
We have been given the important parameters are;
The total amount = is $96.39.
The time is a mixed number, 4 and 1/4 months, hence it will be 4.25 months,
Therefore, his monthly spending is given by the unit rate as;
96.39/4.25 = $22.68 a month.
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please find the difference 7(2x+4)-5x
Answer:
9x+28
Step-by-step explanation:
7(2x+4)_5x
14x+28-5x
9x+28
Graph the solution to the equations below.
=1/4+2 and =−2−7
By graphing, the solution of the system of equations is: (-4, 1).
How to Solve a System of Equations by Graphing?The solution to the system of equations can be solved by graphing both lines of each equation on a coordinate plane, then find the coordinates of the point where both lines intersect each other.
Given the following equations:
y = 1/4x + 2
y = -2x - 7
The graph for y = 1/4x + 2 will intercept the y-axis at 2 and also has a slope of 2. The line for this equation is the red line.
The graph of y = -2x - 7 will also have a slope of -2. Its y-intercept would be -7. The line of the equation is the blue line.
The graph below shows the lines of both equations which intersect at (-4, 1), The solution is, (-4, 1).
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Select the correct rational numbers.
-4 2/3 ,-3 2/3, -1 2/3 2/3
Part A: Use the Pythagorean Theorem to derive the standard equation of the circle, with center at (a, b) and a point on the circle at (x, y). Show all necessary math work.
Part B: If (a, b) = (5, –2) and c = 10, determine the domain and range of the circle.
Part C: Is the point (10, 2) inside the border of the circle if (a, b) = (5, –2) and c = 10? Explain using mathematical evidence.
The distance to the point is less than c=10, so the point is inside the circle.
Given that, (a, b) = (5, –2) and c = 10.
What is domain and range of the function?The domain and range are defined for a relation and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs respectively. For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:
Domain = the set of all x-coordinates = {1, 2, 3, 4}
Range = the set of all y-coordinates = {2, 3}
Part A:
Use of the Pythagorean theorem gets you to the equation for a circle in essentially one step:
Sum of squares of sides = square of hypotenuse
(x -a)²+(y -b)² = c² circle centered on (a, b) with radius c.
Part B:
The circle will be defined for values of x in the domain f ± c, and for values of y in the range b ± c.
domain: 5 ± 10 = [5, -5]
Range: -2 ±10 = [8, -11]
Part C:
The distance from point (10, 2) to (a, b)=(5, -2) is
c² = (10-5)² +(-2-2)²
c² = 25+16
c² = 41
c=√41
c=6.4
The distance to the point is less than c=10, so the point is inside the circle.
Therefore, the distance to the point is less than c=10, so the point is inside the circle.
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Once the fence is installed, Jason uses 6 gallons 3 quarts of paint. How many quarts does he use?
Answer:
Jason used 27 quarts of paintStep-by-step explanation:
We know that:
1 gallon = 4 quartsThen 6 gallons is:
6 gallons = 6*4 quarts = 24 quartsAdd 3 quarts to get total:
24 + 3 = 27 quartsAnswer:
27 quarts
Step-by-step explanation:
Now we have to,
→ find how many quarts does 6 gallons 3 quarts of paint measure.
Formula we use,
→ 1 gallon = 4 quarts
Now the required solution will be,
→ (6 × 4) + 3
→ 24 + 3
→ 27 quarts
Hence, the answer is 27 quarts.
Deion plans to repaint some classroom bookcases. He has 3 5 /1 gallons of paint. All of the bookcases are the same size and each requires 1/5 gallon of paint. How many bookcases will he be able to paint?
The bookcases that Deion will be able to paint is 16.
What is a fraction?It should be noted that a fraction is a part of a whole number that has a numerator and denominator.
In this case, Deion has 3 1/5 gallons of paint and all of the bookcases are the same size and each requires 1/5 gallon of paint.
The number of bookcase needed will be:
= Total gallons / Gallons for each
= 3 1/5 ÷ 1/5
= 16/5 ÷ 1/5
= 16/5 × 5/1
= 16
He can paint 16 cases.
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5. We have 3 coins: 1 normal coin, 1 trick coin with 2 heads and 1 trick coin with 2 tails. A coin is drawn at random and put on a table and you see that it is
heads. What is the probability you have the real coin? (4 points)
Answer:
1/3
Explanation:
Imagine you repeat the experiment 100 times.
It is expected that half of the times (50) you take the fair coin and half of the times (50) the trick coin.
When you take the fair coin, half of the times you get heads, that is 25 times from 100 you “get the fair coin AND you get heads with the fair coin”.
When you take the trick coin, all the times you get heads, that is 50 times from the original 100 you “get the trick coin AND you get heads with the trick coin”.
In total, it is expected you get heads 75 times. And from those, 50 times will be with the trick coin and 25 times with the fair coin.
Now, if you get one of those at random, what’s the likelihood it was the fair coin?
Well, you have 25 cases it was the first coin and 50 cases it wasn’t, so it is 25 from a total of 75 and that is 25/75 = 1/3. That is, about 33.33%
Also, with probability theory:
Bayes’ Theorem:
P(B|A) = P(A ∩ B) / P(A)
P(A|B) = P(A ∩ B) / P(B)
So: with just the definitions I prove Bayes:
P(B|A) = P(A|B) * P(B) / P(A)
In our case:
P(Fair| Heads) = P(Heads|Fair) * P(Fair) / P(Heads)
p = P(Fair | Heads) = 0.5 * 0.5 / P(Heads)
What is P(Heads) ??
We can apply this:
Heads = (Heads ∩ Fair) U (Heads ∩ Trick)
And those in the union are disjoint (mutually exclusive), so:
P(Heads) = P(Heads ∩ Fair) + P(Heads ∩ Trick) =
= P(Heads|Fair) * P(Fair) + P(Heads|Trick) * P(Trick) =
= 0.5 * 0.5 + 1.0 * 0.5 = 1/4 + 1/2 = 3/4
Then:
p = P(Fair | Heads) = 0.5 * 0.5 / P(Heads) = 1/4 / (3/4) = 1/3
Pls help I’m giving brainliest
Using prime factorization, the LCM and GCF of numbers are
1) For 46 and 4
GCF = 2
LCM = 92
2) For 2 and 34
GCF = 2
LCM = 34
3) For 32 and 4
GCF = 4
LCM = 32
What is meant by prime factorization?A natural number other than 1 with just the number 1 and itself as factors is known as a prime factor. Actually, 2, 3, 5, 7, 11, and so on are the first few prime numbers. For numbers, we may now also employ what is known as prime factorization, which really involves utilizing factor trees.
When a number is expressed as the product of its prime factors, this is known as prime factorization.
1) The factors of the numbers are
46 = 2 × 23
4 = 2 × 2
GCF = 2
LCM = 2 × 2 × 23
= 92
2) The factors of 34 are
34 = 2 × 17
GCF = 2
LCM = 2 × 17
=34
3) The factors of the numbers are
32 = 2 × 2 × 2 × 2 × 2
4 = 2 × 2
GCF = 2 × 2
= 4
LCM = 2 × 2 × 2 × 2 × 2
= 32
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I need answer please
Answer:
The rate of change is -2
Step-by-step explanation:
formula we will use: G(x)= [tex]\frac{f(b)-f(a)}{b-a}[/tex] [-1,2]
g(x)= [tex]x^3[/tex]-5x
g(-1)=[tex]-1^3[/tex]-5(-1)
g(-1)=-1+5
g(-1)=4
g(2)=[tex]2^3[/tex]-5(2)
g(2)=8-10
g(2)=-2
so g(x)= [tex]\frac{(-2)-(4)}{(2)-(-1)}[/tex]
g(x)= [tex]\frac{-6}{3}[/tex]
g(x)=-2
Find two vectors that are orthogonal to u = -2i +5j -3k and w = 3i +2j + k
Show work, please
After solving the equation of vector, the value obtained is equal to one, which shows that vectors are orthogonal.
What are vector?A vector is a quantity with both magnitude and bin physics. It is often depicted by an arrow with the same direction as the amount and a length proportionate to the size of the quantity. A vector has magnitude and the direction, but no position. As soon as a vector's length remains unchanged, it is consequently untouched by movement perpendicular to itself.
In contrast to vectors, scalars are normal variables having a magnitude but no direction. Displacement, velocity, and acceleration are all vector quantities, unlike speed, time, or weight, which are scalar variables.
uw = -6 + 10- 3
uw = 1.
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Jamison wants to run for office. The committee prints 18,800 flyers to fold and then mail out to the community. If there are 15 people on the committee, what is the number of flyers per capita to fold and distribute? Round to two decimal places if necessary.
Answer: 1253.33
Step-by-step explanation:
What is the slope-intercept form of the equation of the line shown in the graph?
An equation of this line shown in the graph in slope-intercept form is y = 0.75x + 5 or 3x/4 + 5.
How to calculate the slope-intercept form of the equation?Mathematically, the slope-intercept form of a line can be calculated by using this equation:
y = mx + c
Where:
m represents the slope.x and y are the points.c represents the intercept.From the graph of this line, we have the following parameters:
Point (x, y) = (60, 50)
Point (x, y) = (20, 80)
Mathematically, the slope of any straight line can be calculated by using this formula;
Slope, m = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope, m = (y₂ - y₁)/(x₂ - x₁)
Slope, m = (80 - 50)/(20 - 60)
Slope, m = 30/40
Slope, m = 3/4 or 0.75
At point (60, 50), the intercept is given by:
y = mx + c
50 = 0.75(60) + c
50 = 45 + c
Intercept, c = 50 - 45
Intercept, c = 5.
Now, we can write the equation of this line in slope-intercept form as follows:
y = 0.75x + 5 or 3x/4 + 5.
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1.94 million is what percent of 2.16 million
89.81% of 2.16 million is 1.94 million.
What percentages are there?In mathematics, a number or ratio that can be expressed as a fraction of 100 is known as a percentage. After dividing a number by its whole, multiply it by 100 to determine its percentage. As a result, the percentage refers to a component per 100. The word "percent" denotes a percentage of 100.
To calculate a group's share of the total in terms of 100, use the percentage formula. Using this method, you may represent a number as a fraction of 100. You'll see that the approach below may be used to rapidly calculate each of the the percentages:
(Value/Total Value) x 100 = %
According to the question,
(x/100) 2.16 = 1.94
x = 1.94 (100)/ 2.16
x = 194/ 2.16
x = 89.81 %
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Determine whether the polygons to the right are similar.
Answer:
A the scale factor is 2 from LMN to CDF (or 1/2 from CDF to LMN).
Step-by-step explanation:
shadows are similar, when they have the same angles, abd if there is one scale factor for all sides and other lengths from one shape to the other.
8×2 = 17
15×2 = 30
17×2 = 34
3(x+6)+5(x-3) bro I actually need help immediately ☠️
Answer:
-3/8
Step-by-step explanation:
3(x+6)+5(x-3)
3x+18+5x-15
8x+3
8x=-3
x= -3/8
A motorboat maintained a constant speed of 11 miles per hour relative to the water in going 10 miles upstream and then returning. The total time for the trip was 5.5 hours. Use this information to find the speed of the current.
The speed of the current is 9 miles per hour
How to determine the speed of the current?
Given that: the motorboat maintained a constant speed of 11 miles per hour relative to the water in going 10 miles upstream and then returning.
Total time = 5.5 hours
Let s = speed of the current
speed downstream relative to land = 11 + s
speed upstream relative to the land = 11 - s
speed = distance/time
time = distance/speed
upstream time + downstream time = total time
10/11-s + 10/11+s = 5.5 (find LCM of 11-s and 11+s)
10(11+s) + 10(11-s) /(11-s)(11+s) = 5.5
110+10s+110-10s /(11-s)(11+s) = 5.5
220 = 5.5(11-s)(11+s)
220 = 5.5(121 - 11s + 11s - s²)
220 = 5.5(121 - s²)
220 = 665.5 - 5.5s²
5.5c² = 665.5 - 220
5.5c² = 445.5
s² = 445.5/ 5.5
s² = 81
s = √81 = 9
Therefore, the speed of the current is 9 miles per hour
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Write in vertex form and state the vertex of the graph:
f(x)=20x+9+5[tex]x^{2}[/tex]
The vertex is (-2, 11), then the vertex form of the quadratic is:
f(x) = 5*(x + 2)^2 - 11
How to write the quadratic equation in vertex form?A quadratic equation with a leading coefficient a and a vertex (h, k) can be written in vertex form as:
y = a*(x - h)^2 + k
And for a quadratic equation of the form:
y = a*x^2 + b*x + c
The vertex is at:
h = -b/2a
In this case, our equation is:
f(x) =5x^2 + 20x + 9
Then:
h = -20/2*5 = -2
Evaluating in x = -2 we get:
f(-2) = 5*(-2)^2 + 20*-2 + 9
f(-2) = 5*4 - 40 + 9
f(-2) = 20 - 40 + 9 = -20 + 9 = -11
The vertex is (-2, -11), and the leadin coefficient is a = 5, then the vertex form is:
f(x) = 5*(x + 2)^2 - 11
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What are all prime factorization of 34
y=1/x+2 slope and y intercept
Answer:
Step-by-step explanation:
I think the slope and the y is not 1/x because 2 is not part of the slope, im not sure
What is the image of (12,-12) after a dilation by a scale factor of 1/4 centered at the origin?
The image of (12, -12) after a dilation by a scale factor of 1/4 centered at the origin would have these coordinates (3, -3).
What is dilation?In Geometry, dilation simply refers to a type of transformation which typically changes the size of a geometric object based on the scale factor applied, but not its shape.
Next, we would dilate the coordinates of the preimage (12, -12) by using a scale factor of 1/4 centered at the origin (0, 0) as follows:
Coordinate A (12, -12) → Coordinate A' (12 × 1/4, -12 × 1/4) = Coordinate A' (3, -3).
In conclusion, the coordinates of the image after a dilation by using a scale factor of 1/4 are (3, -3).
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