a. Constant of proportionality is equal to 5.
b. Point (12, 60) represents that 12 pounds of sugar required to make 60 cups of coffee.
What is graph ?The graph is simply a structured representation of the data. It aids in our comprehension of the data. The numerical information gathered through observation is referred to as data.
In a line graph, the information or data is represented as a series of markers, or dots, and is then connected to one another by a straight line.
Usually, data that varies over time is represented with a line graph.
Given:
We have the relation between total number of cups of coffee and the total amount of sugar:
Let y represents the number of cups of coffee
and, x represents the amount of sugar
y ∝ x
y = kx
a. 'k' is the constant of proportionality
When y = 20 , x = 4
20 = k × 4
k = 20/ 4
k = 5
b. From the graph point (12, 60) represents 12 is on the x-axis and y is on the y-axis.
12 represents amount of sugar and 60 represents the number of cups of coffee.
Hence, the Constant of proportionality is equal to 5.
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polynomial function
Answer:
[tex]f(x) = (x+1)(x-2)(x+4)[/tex]
[tex]= (x^{2} +x-2x-2)(x+4)\\\\= (x^{2} -x-2)(x+4)\\\\= x^{3}-x^{2}-2x+4x^{2} -4x-8\\\\= x^{3} +3x^{2} -6x-8[/tex]
If the line in a linear-quadratic system of equations is a vertical line, how many solutions are there?
The number of solutions in the system is 1
How to determine the count of the solutions?From the question, the system of equations is given as
A linear equationA quadratic equationAlso, we understand that the linear equation is a vertical straight line.
So, we have the system to be
A vertical-line linear equationA quadratic equationA vertical line can only intersect with a quadratic line at one point
Hence, there is only one solution in the system
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Determine the convergence of divergence of the sequence. If the sequence converges, find its limit.
The sequence converges and its limit is zero
What is a sequence?A sequence is an ordered set of numbers governed by a rule.
What is the limit of a sequence?The limit of a sequence is the value the sequence approaches
What is convergence of a sequence?The convergence of a sequence is when the sequence approached a definite value
What is divergence of a sequence?The divergence of a sequence is when the sequence does not approach a definite value.
How to determine the convergence of divergence of the sequence?Since we have the sequence (√3 - √1), (√4 - √2), (√5 - √3), (√6 - √2), We need to find the general term of the sequence.
Observing this, we find that the general term Uₙ = √(n + 2) - √n
To determine if the limit converges or diverges, we take the limit of the sequence as n → ∞.
So,
[tex]\lim_{n \to \infty} U_n = \lim_{n \to \infty} \sqrt{n + 2} - \sqrt{n} \\= \lim_{n \to \infty} \sqrt{n + 2} - \lim_{n \to \infty} \sqrt{n} \\= \sqrt{\infty - 2} - \sqrt{\infty} \\= \sqrt{\infty} - \sqrt{\infty} \\= \infty -\infty[/tex]
Since this is an indeterminate form, we use L'hopital's rule to find the limit.
Uₙ = √(n + 2) - √n
= √n{√[(n + 2)/n] - 1}
= √n÷ 1/{√[(1 + 2/n] - 1}
Differentiating, we have
lim n → ∞ Uₙ = lim n → ∞ d√n/dn ÷ d(1/{√[(1 + 2/n] - 1})/dn
= 1/(2√n)/-√[(1 + 2/n] - 1})⁻² × 1/2√(1 + 2/n] × (-2/n²)
= n²[√(1 + 2/n] - 1})²/(2√n)√(1 + 2/n]
= n²[√(1 + 2/n] - 1})²/(2√n)√(1 + 2/n)]
Dividing through by n², we have
= n²[√(1 + 2/n] - 1})²/n² ÷ (2√n)√(1 + 2/n)]/n²
= [√(1 + 2/n] - 1})² ÷ (2(√n)³√(1 + 2/n³)]
Dividing both the numerator and denominator by (√n)³, we have
= [√(1 + 2/n] - 1})²/(√n)³ ÷ (2(√n)³√(1 + 2/n³)]/(√n)³
= (1 + 2/n] - √(1 + 2/n] + 1)/(√n)³ ÷ [2√(1 + 2/n³)]
= (1(√n)³ + 2/n²] - √{(1 + 2/n]/n)³} + 1/(√n)³) ÷ [2√(1 + 2/n³)]
= (1(√n)³ + 2/n²] - √{(1/n³ + 2/n⁴} + 1/√n)³) ÷ [2√(1 + 2/n³)]
So, lim n → ∞ Uₙ = lim n → ∞ (1(√n)³ + 2/n²] - √{(1/n³ + 2/n⁴} + 1/√n)³) ÷ [2√(1 + 2/n³)]
= (1(√∞)³ + 2/∞²] - √{(1/∞³ + 2/∞⁴} + 1/√∞)³) ÷ [2√(1 + 2/∞³)]
= (0 + 0] - √{(0 + 0} + 0) ÷ [2√(1 + 0)]
= (0 - 0 + 0)/2
= 0/2
= 0
The sequence converges and its limit is zero
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Write a recursive formula: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21
Answer:
[tex]a_1=3[/tex]
[tex]a_n=a_n_-_1+2[/tex]
Step-by-step explanation:
Recursive formula : an = a(n-1) + d
where d = common difference , an = nth term, and n = number of terms
Because we are just writing the recursive formula rather than the explicit formula, the only defined term will be "d" or the common difference.
Here, the common difference is +2. This is evident as you must add 2 to get to 5 and then add 2 to get to 7 etc.
So d = 2
Plugging this into the recursive formula we get an = a(n-1) + 2
Also it's very important to state the first term which is 3 so a1 = 3
(x-3)45° find the value of x
Answer:
the value of x is 135 i believe
Step-by-step explanation:
sorry if im wrong
Can I pls get sum help
The values of the variables "a", "b", and "c" are 3.06, 169.74, and 0, respectively. The total amount Gary will pay is $769.74.
Gary purchased a S750 TV on a credit card with a 22% annual percentage rate, and he wants to pay it off in payments of $200 per month. The table shows the information for the first four months after Gary used his credit card. We need to calculate the values of the missing variables and the total amount he will pay.
The variable "a" is the interest charged on the left amount in the third month.
a = (366.68 - 200)*0.018333 = 3.06
The variable "b" is the monthly amount paid by Gary. It is less than or equal to $200, depending on the payment amount. Since the amount to be paid is $169.74, which is less than $200, he will pay the whole amount in one go.
b = 169.74
The variable "c" is the interest charged on the left amount in the fourth month. Since then, the entire amount has been paid.
c = 0
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Stress, measured in newtons per square meter (N/m^2) is the ratio of force, in newtons (N) to the area, in square meters (m^2) on which that force is acting. Based on the table at left, which object is experiencing the greatest stress?
Object: Object 1, Object 2, Object 3, Object 4
Force (N): 72, 185, 406, 512
Area (m^2): 6.46 * 10^-5, 4.56 * 10^-5, 5.07 * 10^-4, 1.14*10^-3
(^Here is the table^)
A) Object 1
B) Object 2 ***** Correct
C) Object 3
D) Object 4
The greatest stress is experienced by object 2.
How to determine the object experiencing the greatest stress?Stress is defined as force divided by area
Given: the table containing values of force and area for objects
Object 1
Stress = 72/6.46×10⁻⁵ = 1114551.084 N/m²
Object 2
Stress = 185/4.56×10⁻⁵ = 4057017.544 N/m²
Object 3
Stress = 406/5.07×10⁻⁴ = 792899.4083 N/m²
Object 4
Stress = 512/1.14×10⁻³ = 449122.807 N/m²
Therefore, object 2 is experiencing the greatest stress
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It takes Oliver 2 hours to wash the windows in his house. It takes his sister 3 hours. Based on this information we can conclude that in one hour, Oliver and his sister would get of the windows washed. This means that it would take hours (which is equal to minutes) for them to get all of the windows washed. Submit Answer attempt 1 out of 5 / problem 1 out of max 1
Answer: 5/6 windows in an hour, 6/5 hours, 72 minutes
Step-by-step explanation: the general formula used is 1/Oliver+1/sister=1/together and you plus in to find the amount of windows in an hour. Once solved, you take the reciprocal of that and that is the hours. Then you solve for the amount of minutes.
construct triangle abc if ac=10cm angle bac=50 degrees and angle bca=50 degrees what type of triangle does this create
Answer:
Angle BAC =50
Step-by-step explanation:
Since O is the circumcenter and angle BOC is the angle subtended by the arc BC at the center and angle BAC is the angle subtended by arc BC
Eva bought seven and one-half pounds of chocolate chips. She used three and three-fourths pounds in some cookies and two and five-sixths pounds in some muffins. How many pounds of chocolate chips were left?
three-twelfths pounds
eleven-twelfths pounds
one and two-twelfths pounds
one and seven-twelfths pounds
The amount of chocolate left is three and seven-twelfths pounds
How to find the amount of chocolate leftThe amount of chocolate left is found by addition and subtraction of fraction
The fraction required are:
Eva bought seven and one-half pounds of chocolate chips = 7 1/2
She used three and three-fourths pounds = 3 3/4
two and five-sixths pounds in some muffins = 2 5/6
total amount used = addition of amount used
3 3/4 + 2 5/6
= 3 11/12
the amount left is gotten by subtraction of amount used from amount Eva bought
= 7 1/2 - 3 11/12
= 3 7/12
this is three seven-twelfths
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Write the equation of a line that has a slope of -3/5 and passes through the point (10,4) in slope intercept form.
[tex](\stackrel{x_1}{10}~,~\stackrel{y_1}{4})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{3}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{- \cfrac{3}{5}}(x-\stackrel{x_1}{10}) \\\\\\ y-4=- \cfrac{3}{5}x+6\implies {\Large \begin{array}{llll} y=- \cfrac{3}{5}x+10 \end{array}}[/tex]
Graph the polygon with vertices A(-3,-1), B(2, 2), C(3,-3) and its image after a rotation 90° about the origin
What are the points???
Geometry chapter 4 review
The image of the polygon after rotation of 90° about the origin is A'(-1, 3) , B'(2 , -2) , C'(-3 , -3) , the graph of the polygon and its image is shown below .
In the question ,
it is given that ,
the vertices of the polygon is A(-3,-1), B(2, 2), C(3,-3) .
the graph of the polygon ABC is shown below .
We know when the point (x , y) is rotated 90° in clockwise direction , it s coordinate change to (y , -x) .
So , image of coordinate A(-3,-1) is
A'(-1,-(-3)) = A'(-1, 3)
image of coordinate B(2, 2) is
B'(2 , -2)
image of coordinate C(3,-3) is
C'(-3 , -3)
So , the graph of the image with coordinates A'(-1, 3) , B'(2 , -2) , C'(-3 , -3) is shown below .
Therefore , The image of the polygon after rotation of 90° about the origin is A'(-1, 3) , B'(2 , -2) , C'(-3 , -3) , the graph of the polygon and its image is shown below .
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Ariel sings a song in 5.92 minutes. Carlos sings the same song in 7.73 minutes. How much longer does it take Carlos to sing the song? Estimate by rounding to the nearest whole number.
HELP
Approximately Carlo takes 2 minutes time to sing the song.
Given that, Ariel sings a song in 5.92 minutes. Carlos sings the same song in 7.73 minutes.
What is the rounding off numbers?Rounding a number means the process of making a number simpler such that its value remains close to what it was. The result obtained after rounding off a number is less accurate, but easier to use. While rounding a number, we consider the place value of digits in a number.
Now,
Difference of singing time = 7.73-5.92
= 1.81 minutes
≈ 2 minutes
Hence, approximately Carlo takes 2 minutes time to sing the song.
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Write the slope-intercept form equation of a line that passes through the points (-1,2) and (4,-8).
Answer: y=-2x
Step-by-step explanation: Use the slope formula
-8-2/4-(-1)
-10/5=-2
y=-2
Answer:
y = - 2x
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (- 1, 2 ) and (x₂, y₂ ) = (4, - 8 )
m = [tex]\frac{-8-2}{4-(-1)}[/tex] = [tex]\frac{-10}{4+1}[/tex] = [tex]\frac{-10}{5}[/tex] = - 2 , then
y = - 2x + c ← is the partial equation
to find c substitute either of the 2 points into the partial equation
using (- 1, 2 )
2 = 2 + c ⇒ c = 2 - 2 = 0
y = - 2x + 0 , that is
y = - 2x ← equation of line
The three angles of a quadrilateral are 60 degrees, 90 degrees and 110 degrees. Determine the fourth angle
Answer:
100°
Step-by-step explanation:
The sum of any polygon angles add to ( n-2)(180) where n is the number of sides ......for a quadrlateral ( 4-2)(180 ) = 360
60 + 90 + 110 + x = 360 shows x = 100 °
Answer:
sin75^°-sin15^°=1/√2
Please help please please
Answer: 44
Step-by-step explanation:
since OPN and RPQ are similar triangles, you can use ratios to find their sides
[tex]\frac{24}{15} =\frac{x}{15+12.5}[/tex] or [tex]\frac{24}{15} =\frac{x}{27.5}[/tex]
cross multiply to get: 24*27.5 = 15x
660/15 = x
x = 44
Nadia is a stockbroker. She earns 13% commission each week. Last week, she sold $7500 worth of stocks. How much did she make last week in commission? If she averages that same amount each week, how much did she
make in commission in 2011?
Answer:6400 multiplied by 13% (0.13) then multiply that answer by 52 , for weeks in the year.
Step-by-step explanation:
6400*.013= $832.00
832*52= $43264.00
waste management company wants to construct a dumpster in the shape of a rectangular solid with no top whose length is four times its width. its volume is fixed at 25.6 yd3. find the dimensions of the dumpster that will minimize its surface area.
the dimensions of the dumpster that will minimize its surface area
length l = 6.32
width w = 1.58
height h = 0.39
Length = l
width = w
height = h
given that length is four times its width
l = 4w
Now, the formula for the surface area of a cube is;
S = 2lh + 2lw + 2wh
substitute l = 4w in the above equation
S = 2(4w)h+2(4w)w+2wh
S = 8wh+8[tex]w^{2}[/tex]+2wh
S = 10wh + 8[tex]w^{2}[/tex]
formula for the volume of a cube is;
V = lwh
now substitute l=4w in the above equation
V = (4w)wh
V = 4[tex]w^{2}[/tex]h
h = 4[tex]w^{2}[/tex]/V
Put V/2w² for h in the surface area equation to get;
S= 8w² + 10w(V/4w²)
= 8w² + 5V/2w
given that volume V = 25.6 substitute that in the above equation
S = 8w² + 5(25.6)/2w
= 8w² + 128/2w
Since we want to minimize the surface area, let us find the first derivative of S to get;
S' = 16w - 128/2w²
At S' = 0
0 = 16w- 128/2w²
16w = 128/2w²
32w³ = 128
w³ = 128/32
w³ = 4
w = [tex]\sqrt[3]{4}[/tex]
w = 1.58 yds
since l = 4w substitute the value of w in that
l = 4(1.58) = 6.32
h = 4[tex]w^{2}[/tex]/V = 4([tex]1.58^{2}[/tex])/25.6
h = 0.39
the dimensions are
length l = 6.32
width w = 1.58
height h = 0.39
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Which of the equations below represent a proportional relationship? Choose all that apply.
Answer:
1
3
6
are all the correct answers because they made the most sense and looked the most proportional
What is the equation of the line that passes through the point (5, 3) and has a slope of 3/5?
Answer:
y -3 = 3/5(x -5) . . . . point-slope formy = 3/5x . . . . . . slope-intercept formStep-by-step explanation:
You want the equation of the line with slope 3/5 that passes through point (5, 3).
Point-slope formThe point-slope form of the equation of a line with slope m through point (h, k) is ...
y -k = m(x -h)
You have m=3/5, (h, k) = (5, 3), so your equation is ...
y -3 = 3/5(x -5) . . . . point-slope form of desired equation
Slope-intercept formThis can be simplified to ...
y -3 = 3/5x -3
y = 3/5x . . . . . . . add 3
a photograph is enlarged into a poster so that the dimensions of the poster are 8 times the dimensions of the photograph. Which statements are true about the perimeter and the area of the poster?
Answer:
I think that the correct answers are C and D. I have questions. how many answers can I choose? what is the original photograph size? don't choose an answer until I get the information that I need
A scale drawing of a rectangular park had a scale of 1 cm = 90 m. 7.6 cm 9.1 cm- What is the actual area of the park in meters squared?
please answer this quickly this is due by 8pm ;((
Step-by-step explanation:
The drawing has the side dimension of 7.6 cm
since each cm represents 90 m the ACTUAL park is
7.6 x 90 = 684 meters
similarly, the bottom of the drawing has the dimension 9.1 cm which represents 9.1 x 90 = 819 m
the area of the park is then
Area = height x width = 684 m x 819 m = 560 196 m^2
OR in one step:
Area of the actual park is then
(7.6 cm * 90 m / cm ) X ( 9.1 cm * 90 m/cm ) = 560 196 m^2
What is the y-intercept, b, of the following linear equation? y = 2x+4 b=
Answer:
4
Step-by-step explanation:
you can use the formula y=mx+b where b is 4
What is the greatest common divisor of x and y?
The only common factor (different than 1) between x and y is 29, so the greatest common divisor is 29.
How to get the greatest common divisor between x and y?
Here we have two real numbers that are written as a product between their corresponding prime factors.
These numbers are:
x = 2*3*5*7*11*13*17*19*23*29
y = 29*31*37*41*43*47*53*59*61*67
Again, all of these are prime numbers, so the only common divisors between the two numbers are the factors that appear in both numbers.
And we can see that the only common factor between x and y is 29, then the greatest common divisor between x and y is 29.
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what is the volume of a trunk of a regular quadrangular pyramid where the side of the base is 18cm, the length of the upper side is 7.5cm and the height of the trunk is 20.4cm
The volume of a trunk of a regular quadrangular pyramid is 2754 cubic centimeters.
According to the question,
We have the following information:
The side of the base is 18cm, the length of the upper side is 7.5cm and the height of the trunk is 20.4cm.
Now, we know that the trunk of a regular quadrangular pyramid is in the shape of a cuboid.
And the following formula is used to find the volume of cuboid:
Length*breadth*height
Volume of a trunk of a regular quadrangular pyramid = 7.5*18*20.4
Volume of a trunk of a regular quadrangular pyramid = 2754 cubic centimeters
Hence, the volume of a trunk of a regular quadrangular pyramid is 2754 cubic centimeters.
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Determine the range of the following graph:
The range of the given function is represented as {y | -5 ≤ y ≤ 5 , y ∈ R} .
Range of a function is defined as the possible values of y or the dependent variable in relation to the values of x.
Here from the graph we can see that the greatest value y achieves is 5.
The lowest value of y from the graph is -5 for all the values of x.
Hence the range will be {-5 ≤ y ≤ 5}
Now the graph is continuous, so we accept all values of y, hence y ∈ R .
Therefore the range will be: {y | -5 ≤ y ≤ 5 , y ∈ R} .
The set of all possible values for the results of the dependent variable, or the entire set of all possible values when the domain is substituted, is the range of a function.
The steps to find the range is:
The range of a function or similar object is the range of all y values, from least to greatest.We determine whether the provided expression of y is positive, negative, or equal to other values by substituting all possible values of x into it.We find the minimum and greatest values of y.To learn more about range visit:
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420.9536 Find the digits in the ones place, in the tenths place, and in the thousandths place for the following number.
Answer:
ones: 0
tenths: 9
thousandths: 3
Step-by-step explanation:
Because 0 is at the beginning of the whole "part" of the number, it is a "one" in place value (confusing right??) for example, in the number 54,743, 3 would be the digit in the ones place.
Because 9 is one decimal place across the example number, it is a tenth - because you would need 10 lots of 0.1 in order to make a whole number.
Because 3 is three decimal places across the example number, it is a thousandth - because you would need 1000 lots of 0.001 to make a whole number.
A good way of remembering these decimal place values is by using the number of zeros after 10, 100, 1,000, 10,000, and so on (one zero is a ten-th, two zeros is a hundred-th, etc, etc.)
6x+22=13x-41
1234567890
Answer:
x=9
Step-by-step explanation:
I assume you want me to solve for x.
6x+22=13x-41
Subtract 6x from both sides.
22=7x-41
Add 41 to both sides.
63=7x
Divide both sides by 7.
x=9.
CHECK:
6x+22=54+22=76
13x-41=117-41=76
76=76
So, x=9
-3+-u=-4 Please help!!!!!
A rectangular swimming is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a consistent 2 feet wide and has as area of 196 square feet. What are the dimensions of the pool? If the depth of the pool is 6 feet, how much water can the pool hold if the height of water is 1 foot below the top of the pool?
Convert cubic feet to gallons
The length of the swimming pool is 30 feet
The width of the swimming pool is 15 feet
The volume of the swimming pool is 16875 gallons
Given,
Length of the rectangular swimming pool = 2x
Width of the rectangular swimming pool = x
A small concrete walkway surrounds the pool.
The width of the walkway = 2 feet
The area of the walkway = 196 square feet
We have to find the dimensions of the pool;
Here,
Area of rectangle = length x width
Area of walkways = (4x + 8x) + 4 x 4 = 196
12x + 16 = 196
12x = 196 - 16
12x = 180
x = 180/12
x = 15
That is,
Width of the pool is 15 feet
Length of the pool = 2 × x = 15 x 2 = 30 feet
Now,
If the depth of the pool is 6 feet, we have to find the amount of water that the pool can hold if the height of water is 1 foot below the top of the pool.
That is,
Volume of pool = length x width x depth
Here, depth = 6 - 1 = 5 feet
So,
Volume = 15 x 30 x 5 = 2250 cubic feet
Convert cubic feet to gallons
2250 x 7.5 = 16875 gallons of holding capacity
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