A cafeteria serves lemonade that is made from a powdered drink mix. There is a proportional relationship between the number of scoops of powdered drink mix and the amount of water needed to make it. For every 2 scoops of mix, one-half gallon of water is needed, and for every 6 scoops of mix, one and one-half gallons of water are needed.
Part A: Find the constant of proportionality. Show every step of your work. (4 points)
Part B: Write an equation that represents the relationship. Show every step of your work. (2 points)
Part C: Describe how you would graph the relationship. Use complete sentences. (4 points)
Part D: How many gallons of water are needed for 10 scoops of drink mix? (2 points)
please help asap its almost to late
All the answers to the given parts are mentioned below -
What is the general equation of a Straight line? How it represents a proportional relationship?The general equation of a straight line is -
[y] = [m]x + [c]
where -
[m] is slope of line which tells the unit rate of change of [y] with respect to [x].
[c] is the y - intercept i.e. the point where the graph cuts the [y] axis.
y = mx also represents direct proportionality. We can write [m] as -
m = y/x
OR
y₁/x₁ = y₂/x₂
We have a cafeteria serves lemonade that is made from a powdered drink mix. There is a proportional relationship between the number of scoops of powdered drink mix and the amount of water needed to make it. For every 2 scoops of mix, one-half gallon of water is needed, and for every 6 scoops of mix, one and one-half gallons of water are needed.
We can write the proportional relationship as -
y = kx
Now, from the given information, we can write -
2 scoops need 0.5 gallon of water
6 scoops need 1.5 gallon of water
So -
k = 2/0.5
k = 2/(1/2)
k = 2 x 2 = 4
Equations that represents the relationship can be written as -
y = 4x + c
Now, 2 scoops need 0.5 gallon of water.
2 = 4 x 1/2 + c
2 = 2 + c
c = 0
So, the equation will be y = 4x.
Graph of y = 4x is attached at the end.
For 10 scoops of water -
10 = 4x
x = 2.5 gallons of water is needed.
Therefore, all the answers to the given parts are mentioned above.
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Algebraic models grade 12 math please write the answers without explaining thank you.
Explanation
let's remember this property of the exponent number
[tex]a^m\cdot a^n=a^{m+n}[/tex]Step 1
solve by applying the property. ( let the same base and add the exponents)
[tex]\begin{gathered} 6^4\cdot6^{-5} \\ 6^4\cdot6^{-5}=6^{4+(-5)} \\ 6^4\cdot6^{-5}=6^{-1} \\ \end{gathered}[/tex]hence, the answer is
[tex]d)6^{-1}[/tex]I hope this helps you
What is the end behavior of f(x) =2^–x− 5 as x goes to infinity?
Given the function:
[tex]f(x)=2^{-x}-5[/tex]As x goes to infinity, the term (2^-x) will go to zero
And the overall value of the function will be -5
So, the end behavior of f(x) as x goes to infinity = -5
A book sold 33,400 coples in its first month of release. Suppose this represents 7.6% of the number of coples sold to date. How many coples have been sold todate?Round your answer to the nearest whole number.
The number sold in the first month is given as 33,400.
This number is 7.6 percent of the total copies sold till date. This means x copies have been sold till date, and x copies represents 100 percent.
Therefore, you would have the following proportion;
[tex]\begin{gathered} \frac{33400}{x}=\frac{7.6}{100} \\ \text{Cross multiply and you'll have;} \\ \frac{33400\times100}{7.6}=x \\ 439473.684210\ldots=x \\ x\approx439474\text{ (rounded to the nearest whole number)} \end{gathered}[/tex]The number of copies sold till date is 439,474 (rounded to the nearest whole number)
Jessica is deciding on her schedule for next semester. She must take each of the following classes: English 101, Spanish 102, Biology 102, andCollege Algebra. If there are 15 sections of English 101,9 sections of Spanish 102, 13 sections of Biology 102, and 15 sections of College Algebra,how many different possible schedules arethere for Jessica to choose from? Assume there are no time conflicts between the different classes.Keypad
Jessica must take four classes: English, Spanish, Biology, and College Algebra.
There are:
15 sections of English
9 sections of Spanish
13 sections of Biology
15 sections of College Algebra.
She has 15 possible choices for English class. Once selected, she has 9 choices for Spanish class.
There is a total of 15*9 = 135 possible schedules for both subjects.
When we combine this with the rest of the classes, we find a total of:
15*9*13*15 = 26,325 possible schedules, assuming there are no time conflicts between them.
Answer: 26,325
A cookie recipe called for 3 ¼ cups of sugar for every 2 ⅓ cups of flour. If you made a batch of cookies using 4 cups of flour, how many cups of sugar would you need?
1) Gathering the data
3 ¼ cups of sugar------------------ 2 ⅓ cups of flour
x 4
2) Let's set a proportion, and then cross multiply those ratios but before that
let's convert those mixed numbers:
[tex]\begin{gathered} 3\frac{1}{4}=\frac{4\times3+1}{4}=\frac{13}{4} \\ 2\frac{1}{3}=\frac{3\times2+1}{3}=\frac{7}{3} \end{gathered}[/tex][tex]\begin{gathered} \frac{13}{4}-----\frac{7}{3} \\ x\text{ -------4} \\ \frac{7}{3}x=4\times\frac{13}{4} \\ \frac{7}{3}x=13 \\ 7x=39 \\ x=\frac{39}{7} \end{gathered}[/tex]So rewriting it above, we have. 39/7 as 39/7 is >1 then we can rewrite it into a Mixed Number:
3) Hence, I'll need 5 4/7 cups of sugar
On which number line the location of point P represent the probability of an event that is likely, but not certain?
The straight line that best represents something probable but not certain is option D.
It shows a probability of approximately 80%.
How do you write this as a power? 8x8x8x8x8x8 A) 86 B) 86 C) 68 D) 87
8 x 8 x8 x8 x8 x8 can be written as;
[tex]8^6[/tex]The correct option is A) [tex]8^6[/tex] . In the expression 8x8x8x8x8x8 can be written in the exponentiation as [tex]8^6[/tex].
.
To write the expression 8x8x8x8x8x8 as a power, we can use exponentiation to represent the repeated multiplication.
Step by step, we can simplify the expression as follows:
Start with the given expression: 8x8x8x8x8x8.
Since all the factors are the same (8), we can rewrite the expression using the base (8) and the exponent representing the number of times it is multiplied (6 times).
Therefore, we can write 8x8x8x8x8x8 as [tex]8^6[/tex].
Hence, the expression 8x8x8x8x8x8 can be written in the exponentiation as [tex]8^6[/tex]. The correct option is A) [tex]8^6[/tex]
.
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solve using elimination-10x - 10y = 10 -2x - 5y = -19(_ , _)
The given system of equations is expressed as
-10x - 10y = 10
-2x - 5y = -19
The first step is to make the coefficients of one of the variables in each equation equal. To make the coefficient of x equal, we would multiply the first equation by 2 and the second equation by 10. It becomes
- 20x - 20y = 20 equation 3
- 20x - 50y = - 190 equation 4
We would eliminate x by subtracting equation 4 from equation 3. It becomes
- 20x - - 20x - 20y - - 50y = 20 - - 190
- 20x + 20x - 20y + 50y = 210
30y = 210
y = 210/30
y = 7
Substituting y = 7 into the second equation, it becomes
- 2x - 5 * 7 = - 19
- 2x - 35 = - 19
- 2x = - 19 + 35 = 16
x = 16/- 2
x = - 8
The solution is
(- 8, - 7)
Select ALL the pairs of points so that the line between these points has a slope of 2/3?(0,0) and (3, 2)(1,5) and (4,7)(-2,-2) and (4,2)0 (0,0) and (2,3)(20, 30) and (-20, -30)
Answer:
• (0,0) and (3, 2)
,• (1,5) and (4,7)
,• (-2,-2) and (4,2)
Explanation:
Points (0,0) and (3, 2)
[tex]m=\frac{2-0}{3-0}=\frac{2}{3}[/tex]Points (1,5) and (4,7)
[tex]m=\frac{7-5}{4-1}=\frac{2}{3}[/tex]Points (-2,-2) and (4,2)
[tex]m=\frac{2-(-2)}{4-(-2)}=\frac{2+2}{4+2}=\frac{4}{6}=\frac{2}{3}[/tex]Points (0,0) and (2,3)
[tex]m=\frac{3-0}{2-0}=\frac{3}{2}[/tex]Points (20, 30) and (-20, -30)
[tex]m=\frac{30-(-30)}{20-(-20)}=\frac{60}{40}=\frac{3}{2}[/tex]The first three options are correct.
write an equation of the circle that passes through (2, 8) with center (-3 4)
we have, the equation is of the form
[tex](x-h)^2+(y-k)^2=r^2[/tex]then, first calculate the radius of the circle
[tex]\begin{gathered} r=\sqrt[]{(x2-x1)^2+(y2-y1)} \\ r=\sqrt[]{(14-14)^2+(-8-1)^2} \\ r=\sqrt[]{0+(-9)^2} \\ r=\sqrt[]{81} \\ r=9 \end{gathered}[/tex]so, (h,k) is the center and the equation is
[tex]\begin{gathered} (x-14)^2+(y-(-8))^2=9^2 \\ (x-14)^2+(y+8)^2=81 \end{gathered}[/tex]Campsite A and campsite B are located directlyopposite each other on the shores of Lake Omega,as shown in the diagram below. The two campsitesform a right triangle with Sam's position, S. Thedistance from campsite B to Sam's position is1,300 yards, and campsite A is 1,700 yards from hisposition1,700 yardsLake OmegaB1,300 yardsSWhat is the distance from campsite A to campsiteB, to the nearest yard?
Answer:
The distance from campsite A to campsite B is;
[tex]1095\text{ yards}[/tex]Explanation:
Given that The two campsites A and B form a right triangle with Sam's position, S.
The distance from campsite B to Sam's position is 1,300 yards, and campsite A is 1,700 yards from his position;
[tex]\begin{gathered} AS=1700\text{ yards} \\ BS=1300\text{ yards} \end{gathered}[/tex]Applying pythagorean theorem, to solve for the third side AB;
[tex]\begin{gathered} AS^2=AB^2+BS^2 \\ \text{making AB the subject of formula;} \\ AB^2=AS^2-BS^2 \\ AB=\sqrt[]{AS^2-BS^2} \end{gathered}[/tex]Substituting the given values;
[tex]\begin{gathered} AB=\sqrt[]{1700^2-1300^2} \\ AB=\sqrt[]{1200000} \\ AB=1095\text{ yards} \end{gathered}[/tex]Therefore, the distance from campsite A to campsite B is;
[tex]1095\text{ yards}[/tex]2x-5y= -19
-3x+y=9
solve by substitution
Answer: (-2,3)
Step-by-step explanation:
2x-5y=-19 (1)
-3x+y=9 (2)
2x-5y=-19 (3)
y=3x+9 (4)
2x-5(3x+9)=-19
2x-15x-45=-19
-13x=-19+45
-13x=26
Divide both parts of the equation by -13:
x=-2
Substitute the value of x=-2 into equation (4):
y=3(-2)+9
y=-6+9
y=3
Thus, (-2,3)
what is the size of rectungle 2x2x2
Perimeter: 8 u
Area: 4 u^2
Volume: 8 u^3
Explanation:
u = unit (cm / m etc...)
side = 2 u
Formula for a rectangle:
Perimiter : 2*(side + side)
=> 2 * ( 2 + 2) = 8 u
Area: side * side
=> 2 * 2 = 4 u^2
Volume: Area * Height
=> 2 * 4 u^2 = 8 u^2
Help fast plssssss!!
Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 16 gallons of fuel, the airplane 2104. When carrying 48 gallons of fuel, it weighs 2312. How much does the airplane weigh if it is carrying 66 gallons of fuel?
Let 'y' be the weight of the airplane corresponding to when the amount of fuel is 'x'.
Given that the airplane weighs 2104 when fuel is 16 gallons, this can be represented as the ordered pair,
[tex](16,2104)[/tex]Also, given that the airplane weighs 2312 when fuel is 48 gallons.The corresponding ordered pair will be,
[tex](48,2312)[/tex]It is mentioned that there is a linear relationship between the amount of fuel (x), and the weight of airplane (y).
Consider that the equation of a straight line passing through two given points is given by,
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}\cdot(x_2-x_1)[/tex]As per the given problem,
[tex]\begin{gathered} (x_1,y_1)=(16,2104) \\ (x_2,y_2)=(48,2312) \end{gathered}[/tex]Substitute the values,
[tex]\begin{gathered} y-2104_{}=\frac{2312-2104}{48-16}\cdot(x-16) \\ y-2104_{}=\frac{13}{2}\cdot(x-16) \\ y-2104=\frac{13}{2}x-104 \\ y=\frac{13}{2}x-104+2104 \\ y=\frac{13}{2}x+2000 \end{gathered}[/tex]At the instant when the fuel is 66 gallons,
[tex]x=66[/tex]The corresponding weight of airplane is calculated as,
[tex]\begin{gathered} y=\frac{13}{2}(66)+2000 \\ y=429+2000 \\ y=2429 \end{gathered}[/tex]Thus, the airplane weighs 2429 if it is carrying 66 gallons of fuel.
answer.The number of cities in a region over time is represented by the function C(=) = 2.9(1.05). The approximate number of people per city isrepresented by the function P(t) = (1.05)35 +5.Which function best describes T(*), the approximate population in the region?OA T(I) = (3.045)* + (1.05)35 +5OB. T(1) = (6.09)45+5OC. T() = 2.9(1.05)45+5OD. Т(1) = 2.9(1.05)352 +55
Given:
[tex]\begin{gathered} \text{Number of cities: }C(x)=2.9(1.05)^x \\ \\ \text{Number of people per city: P}(x)=(1.05)^{3x+5} \end{gathered}[/tex]Let's solve for T(x) which represents the approximate population in the region.
To find the approximate population in the region, apply the formula:
[tex]T(x)=C(x)\ast P(x)[/tex]Thus, we have:
[tex]T(x)=2.9(1.05)^x\ast(1.05)^{3x+5}^{}[/tex]Let's solve the equation for T(x).
Thus, we have:
[tex]\begin{gathered} T(x)=2.9((1.05)^{3x+5}(1.05)^x) \\ \\ Apply\text{ power rule:} \\ T(x)=2.9(1.05)^{3x+5+x^{}_{}} \\ \\ T(x)=2.9(1.05)^{3x+x+5} \\ \\ T(x)=2.9(1.05)^{4x+5} \end{gathered}[/tex]Therefore, the function that best describes the approximate population in the region is:
[tex]T(x)=2.9(1.05)^{4x+5}[/tex]ANSWER:
C
[tex]T(x)=2.9(1.05)^{4x+5}[/tex]101987006HR5SrOL4.3ON21.1 2 345 6 78 9 10Which of these statements are true for the scatter plot? Select all that apply.The scatter plot shows a negative association.The scatter plot shows a linear association.The scatter plot shows a positive association.The scatter plot shows no association.
The scatter plot shows a linear association.
The scatter plot shows a positive association.
ty-qy+p=r solve for y
factor they expression completely 9x−21
Answer:
3(3x - 7)
Step-by-step explanation:
9x - 21
GCF of 9 and 21 is 3
3(3x - 7)
I hope this helps!
a school ordered three large boxes of board markers after giving 15 markers to each of three teachers there were ninety X the diagram represents the situation how many markers were original in the
Determine the value of x.
[tex]\begin{gathered} x-15+x-15+x-15=90 \\ 3x=90+45 \\ x=\frac{135}{3} \\ =45 \end{gathered}[/tex]So there are 45 markers originally in each box.
Write 6.546 x 10 ^-6 in standard notation
The given number in scientific notation is:
[tex]6.546\times10^{-6}[/tex]In order to write it in standard notation, first take a look at the power: -6.
As it is a negative number, it means we need to move the dot to the left 6 units.
Therefore, we need to fill the blank places with zeros:
[tex]6.546\times10^{-6}=0.000006546[/tex]through: (5, 5), slope = 10
Answer: The correct answer is y = 10x – 45
Step-by-step explanation:
When graphing a line with a slope of 10 from point (5,5), we find that the y-intercept (where the line crosses the y-axis) is -45
Use the slope-intercept form (y=mx+b), where m=slope (10) and b=the y-intercept (-45)
y = 10x - 45
A rectangular pyramid has a volume of 90 cubic feet. What is the volume of a rectangular prism with the same size base and same height?choice;45 cubic feet90 cubic feet270 cubic feet30 cubic feet
Solution
Step-by-step explanation:
Here we are given the volume of rectangular pyramid as 90 cubic feet as we are required to find the volume of rectangular prism.
For that we need to use the theorem which says that
the volume prism is always one third of the volume of the pyramid . Whether it is rectangular of triangular base. Hence in this case also the volume of the rectangular prism will be one third of the volume of the rectangular pyramid.
Volume of Rectangular prism = 1/3 x Volume of rectangular pyramid
[tex]\begin{gathered} \frac{1}{3}\times90 \\ =30\text{ cubic feet} \end{gathered}[/tex]Therefore the volume of the rectangular pyramid = 30 cubic feet
A robot can complete 10 tasks in hour. Answer in fraction form.How many tasks can the robot complete in 1 hour?How long does it take the robot to complete one task?hoursIf you have a mixed number put a space between the whole number and fraction. Ex 7=7 1/2tasks
Hello!
We know that the robot can complete 10 tasks in 3/4 hour.
To solve this exercise, we can use the rule of three, look:
How many tasks can the robot complete in 1 hour?[tex]\frac{\frac{3}{4}h\rightarrow10\text{ tasks}}{1h\rightarrow x\text{ tasks}}[/tex]So, we have:
[tex]\begin{gathered} \frac{3}{4}x=10 \\ \\ x=\frac{10}{1}\div\frac{3}{4}\rightarrow\frac{10}{1}\times\frac{4}{3}=\frac{40}{3} \\ \\ \end{gathered}[/tex]Let's write it as a mixed number:
Answer:
[tex]13\text{ }\frac{1}{3}\text{ tasks}[/tex]How long does it take the robot to complete one task?We'll solve it in a similar way, look:
[tex]\frac{\frac{3}{4}h\operatorname{\rightarrow}10\text{ tasks}}{x\text{ }h\operatorname{\rightarrow}1\text{ task}}[/tex][tex]\begin{gathered} 10x=\frac{3}{4} \\ \\ x=\frac{3}{4}\div\frac{10}{1}\rightarrow\frac{3}{4}\times\frac{1}{10}=\frac{3}{40}\text{ hours} \end{gathered}[/tex]Answer:
[tex]\frac{3}{40}\text{ hours}[/tex]find a slope of the line that passes through (8,2) and (6,3)
EXPLANATION
Given the dots:
(x1,y1)=(8,2) and (x2,y2)=(6,3)
The slope equation is:
[tex]\text{Slope = }\frac{(y_2-y_1)}{(x_2-x_1)}[/tex]Replacing the ordered pairs in the slope equation will give us:
[tex]\text{Slope = }\frac{(3-2)}{(6-8)}=\frac{1}{-2}=-\frac{1}{2}[/tex]The slope of the line is -1/2.
A bag contains 50 marbles. Marsha chooses a marble, records its color, and then puts the marble back into the bag. Marsha repeats this process and records 7 red marbles and 3 yellow marbles. What is the ratio of the number of red marbles to the number of yellow marbles she chooses
Probability is the chance that an event will occur and it's measured by the ratio of the favorable cases to the total number of cases possible.
The ratio for probability is given by:
[tex]\begin{gathered} \text{ratio of 3 marbles:ratio of yellow marbles} \\ 7\colon3 \end{gathered}[/tex]F(x)=2|x-1| Graph using transformations and describe the transformations of the parent function y =x^2.
[Please see that in the question should be a mistake regarding the parent function. It should be written y = |x| instead of y = x².]
To answer this question, we need to know that the below function is a transformation of the parent function, f(x) = |x|:
[tex]f(x)=2|x-1|[/tex]Describing the transformationsTo end up with the above function from the parent function, we need to follow the next steps:
1. Translate the function, y = |x| one unit to right. We can do this by subtracting one unit to the parent function as follows:
[tex]f(x)=|x-1|[/tex]We can see this graphically as follows:
The blue function is the first transformation of the parent function, f(x) = |x|.
2. The function has been dilated by a factor of 2 from the x-axis. That is, the function has been dilated by a factor of 2 vertically. Then, we have:
[tex]f(x)=2|x-1|[/tex]And now, we can see the transformation graphically as follows:
Therefore, the blue line is the graph representation of the function:
[tex]f(x)=2|x-1|[/tex]determine if each graph compares the diameter and the circle with the circle's radius area or circumference
The radius of the circle is half of the diameter. Therefore, if the diameter is 2 units, then the radius is 1 unit. If the diameter is 6 units, the radius will be 3 units. The graph that represents the relationship between radius and diameter is Graph B.
The circumference of the circle can be solved by multiplying the diameter and the value of pi. Therefore, this is a linear function. If the diameter is 4 units, the circumference is approximately 12.57 units. If the diameter is 6 units, the circumference is approximately 18.85 units. The graph that best represents the relationship between diameter and circumference is Graph C.
Lastly, the area of the circle with respect to the diameter is a quadratic function due to the nature of the formula that is A = πr². The graph of a quadratic function is parabolic in nature. Therefore, the graph that best represents the relationship between diameter and area is Graph A.
To summarize, the vertical axis for each graph is:
Graph A → Area
Graph B → Radius
Graph C → Circumference
Emmet opened a savings account and deposited 1,000.00 as principal the account earns 8%interest compounded monthly what is the balance after 9 years
We have to use the compound interest formula to solve this problem.
The compound interest formula:
[tex]F=P(1+r)^n[/tex]Where
F is the future value [what we are solving for]
P is the principal, or initial, amount [It is $1000]
r is the rate of interest per period [It is given 8% annual interest, so 8/12 = 0.66% per month, in decimal that is r = 0.0066]
n is the time period [monthly compounding for 9 years is n = 12 * 9 = 108]
Now, we can substitute all the known information and solve for F:
[tex]\begin{gathered} F=P(1+r)^n^{} \\ F=1000(1+0.0066)^{108} \\ F=2048.06 \end{gathered}[/tex]After 9 years, the balance is:
$2048.06