Answer:
C = (7, 6)
Step-by-step explanation:
The problem statement tells us the relation between the points is ...
(B-A)/(C-A) = 2/7
7(B -A) = 2(C -A) . . . . . . multiply by 7(C-A)
7B -7A +2A = 2C . . . . . add 2A
C = (7B -5A)/2 . . . . . . . divide by 2
C = (7(2, -4) -5(0, -8))/2 = (14, 12)/2 . . . . . fill in the values of A and B
C = (7, 6)
For Emily's birthday, her father treated her and five friends to a dinner at a restaurant at the mall. Emily and her friends ordered three pizza dinners at $6.25 each and three chicken baskets at $7.50 each. All six people ordered soft drinks costing $1.25 each. The sales tax on the meal was 4.5%, and the tip was 15%. What was the total cost of the meal, including the sales tax and the tip?
Answer:
$58.26.
Step-by-step explanation:
Total cost without sales tax and tip = 3 * 6.25 + 3 * 7.50 + 6 *1.25
= $48.75
Plus Sales tax and tip:
= 48.75 + 0.045*48.74 + 0.15*48.75
= $58.26.
Answer:
58.25
Step-by-step explanation:
How would you find the volume of a tower created from 1,000 cans that were each 12oz in volume?
Answer:
Multiply the number of cans by the volume of each: 12,000 oz.
Step-by-step explanation:
You find the total volume of more than one can by adding the volumes of the cans involved.
For 2 cans, the volume would be ...
12 oz + 12 oz = 24 oz
__
When you consider adding numbers more than a couple of times, you start looking for ways to simplify the effort. Multiplication was invented for that purpose. Here, multiplying the volume of 1 can by 1000 is the same as adding the volumes of 1000 cans.
For 1000 cans with volume of 12 oz each, the volume of the total is ...
1000 × 12 oz = 12,000 oz.
solve the given equation and check the solution 7/2x - 5/2 = 23/2
Answer:
The X, should be 4
Step-by-step explanation:
7/2x - 5/2 = 23/2
7/2x = 23/2 + 5/2
7/2 = 14
x = 14 . 2/7
x= 4
The solution to the equation [tex]\frac{7}{2}x - \frac{5}{2} = \frac{23}{2}[/tex] is x = 4. By substituting x = 4 back into the equation, we confirm that it satisfies the original equation, proving that the solution is correct.
Let’s solve the equation [tex]\frac{7}{2}x - \frac{5}{2} = \frac{23}{2}[/tex] step-by-step as shown below-
Let's isolate the x-term by adding [tex]\frac{5}{2}[/tex] to both sides of the equation
[tex]\frac{7}{2}x - \frac{5}{2} + \frac{5}{2} = \frac{23}{2} + \frac{5}{2}[/tex]This simplifies to:
[tex]\frac{7}{2}x = \frac{28}{2}[/tex]Next, simplify the right-hand side:
[tex]\frac{7}{2}x = 14[/tex]Now, multiply both sides by [tex]\frac{7}{2}[/tex] to solve for x:
[tex]x = \frac{(14 \times 2)}{7}[/tex][tex]x = \frac{28}{7}[/tex]And the final solution is:
[tex]x = 4[/tex]To check the solution, substitute x = 4 back into the original equation:
[tex]\frac{7}{2} \times 4 - \frac{5}{2} = \frac{23}{2}[/tex]This becomes:
[tex]14 - \frac{5}{2} = \frac{23}{2}[/tex]The left side simplifies to:
[tex]\frac{28}{2} - \frac{5}{2} = \frac{23}{2}[/tex]Which confirms:
[tex]\frac{23}{2} = \frac{23}{2}[/tex]Thus, the solution x = 4 is correct.
what is the range of the following set? (-2,4), (0,3), (1,6), (-1,2)
a. (4,0,6,-1)
b. (-2,0,1,-1)
c. (-2,4,0,3,1,6,-1,2)
d. (4,3,6,2)
e. (-2,3,1,2)
Answer:
d. (4,3,6,2)
Step-by-step explanation:
The list of second numbers of the ordered pairs is the range.
___
The second number of the first pair is 4; the second number of the second pair is 3. The only 4-number answer choice containing these two numbers is (d). You don't even have to work the whole problem to make the proper choice.
List S and list T each contain 5 positive integers, and for each list the average (arithmetic mean) of the integers in the list is 40. If the integers 30, 40, and 50 are in both lists, is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T? (1) The integer 25 is in list S. (2) The integer 45 is in list T.
Answer:
Yes, SDS > SDT
Step-by-step explanation:
List S
25, 30, 40, 50, XS
Average list S = 40
So, we could write,
Average list S = 40 = (25 + 30 + 40 + 50 + XS) / 5
Solving for XS
XS = 40 x 5 – 25 – 30 – 40 – 50 = 200 – 145 = 55
SDs = SD (25, 30, 40, 50, 55) = 12.74
List T
30, 40, 45, 50, XT
Average list T = 40
So, we could write,
Average list T = 40 = (30 + 40 + 45 + 50 + XT) / 5
Solving for XT
XT = 40 x 5 – 30 – 40 – 45 – 50 = 200 – 165 = 35
SDT = SD (30, 35, 40, 45, 50) = 7.1
Even at first sight SDS > SDT because 25 is out of the range 30-50, while 45 is within that range.
List S is more spread than list T.
A lower or higher number than the mean in a list can increase the standard deviation. In this case, the standard deviation of list S is greater than list T due to the presence of 25, which is farther from the mean of 40 than the numbers in list T.
Explanation:The question asked relates to the standard deviation of two sets of positive integers, list S and list T. For each list, we calculate the standard deviation, which is a measure of how spread out the numbers are around the mean value. Based on the information provided:
The integer 25 is in list S. The integer 45 is in list T.The presence of a number lower or higher than the mean in list S or T respectively, will increase the standard deviation because the deviation or difference from the mean is greater. Remember, standard deviation measures the variation or dispersion from the average. Hence, if the integer 25 is in List S and the integer 45 is in list T, the standard deviation will be greater for list, S, considering these numbers are farther from the mean of 40 compared to the numbers in List T.
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I need some help with this problem
Answer:
1. No
2.Yes
3.Yes
4.Yes
5.No
Step-by-step explanation:
Plug in the numbers
1. 3+3 = 9+9 / 6 = 18 / NO
2. 20-0 = 20+0 / 20 = 20 / Yes
3. 3+0/3-0 / 3/3 = 1 / Yes
4. 4+3 = 16-9 / 7 = 7 / Yes
5. 0+1 = 1-0/0 / 1 = 1/0 / No
Rate as Brainliest plz
Answer: No Yes Yes Yes No
Step-by-step explanation: brain power
A geyser Erupts every fourth day . Another geyser erupts every sixth day. Today both geysers erupted. In how many days will both geysers erupt on the same day again?
In 12 days both geysers erupt on the same day again
What is Least common multiple?The smallest number that is a multiple of each of two or more numbers.
Given:
A geyser Erupts every fourth day.
Another geyser erupts every sixth day.
so, to find how many days will both geysers erupt on the same day again
we have to find the LCM of 4 and 6
So, 4 = 2*2
6= 2*3
LCM (4, 6) =2*2*3 = 12
Hence, 12 days both geysers erupt on the same day again.
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Tessa made a mistake solving the equation 4(2x + 3 ) = -20. Select the line in which her mistake appears.
Line 1. 4 (2x + 3) = -20
Line 2. 8x + 12 = -20
Line 3. 8x = -8
Line 4. x = -1
Answer:
Line 3
Step-by-step explanation:
we have
[tex]4(2x+3)=-20[/tex]
Verify each line
Line 1
[tex]4(2x+3)=-20[/tex] ----> the given equation
Line 1 is correct
Line 2
Distribute in the left side
[tex]8x+12=-20[/tex]
Line 2 is correct
Line 3
Subtract 12 both sides
[tex]8x=-20-12[/tex]
[tex]8x=-32[/tex]
The line 3 is not correct
Tessa made a mistake in Line 3 of the problem. The correct calculation should be -20 minus 12, which equals -32. The solution for x is -4.
Explanation:The subject of this problem is Mathematics, specifically algebra. Looking at the lines of the problem you provided, we can see that Tessa's mistake lies in Line 3 when she subtracts 12 from -20 and gets -8. The correct subtraction should be -20 minus 12 = -32.
So, Line 3 would correctly be 8x = -32 . To solve for x, we would then divide -32 by 8 to get x = -4.
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A women's hospital reported 212 deliveries during June. Two sets of twins were born. There were 215 obstetrical discharges; 214 births; four women had first-time C-sections; and three women had a repeat C-section. The C-section rate for September is 3.30 percent. True or false?
Answer:
Impossible to know. Numbers provided on the statement are for June, though question is made for september. If, by any chance, question is not correctly stated, answer will be True
Step-by-step explanation:
According to the problem, there were 214 births. Seven of those 214 births were via C-section. Then: 100%*(7/214)=3.3%
It does not matter which births were natural, or C sectioned, as they are considered in the same group.
Suppose that, in some distant part of the universe, there is a star with four orbiting planets . One planet makes a trip around the star in 6 earth years , the second planet takes 9 earth years, the third takes 15 earth years and the fourth takes 18 earth years . Suppose that at some time the planets are lined up. How many years will it take for them to all line up
Answer: 90 Earth years.
Step-by-step explanation:
Analizing the information provided in the exercise, you need to find the Least Common Multiple (LCM) of the given numbers.
You can follow these steps:
1. You must descompose 6, 9, 15 and 18 into their prime factors:
[tex]6=2*3\\\\9=3*3=3^2\\\\15=3*5\\\\18=2*3*3=2*3^2[/tex]
2. Finally, you need to choose the commons and non commons with their greatest exponents and multiply them. Then you get:
[tex]L.C.M=2*3^2*5=2*9*5\\\\L.C.M=90[/tex]
Therefore, it will take 90 Earth years for them to all line up.
At Central Online High School, 4510045100 of the students have a dog, 3010030100 have a cat, and 1810018100 have both a dog and a cat. What is the probability that a student who has a dog also has a cat? Enter your answer as a reduced fraction with the / symbol, like this: 3/14
Answer: [tex]\dfrac{2}{5}[/tex]
Step-by-step explanation:
Given : The proportion of students have a dog : [tex]P(D)=\dfrac{45}{100}[/tex]
The proportion of students have a cat : [tex]P(C)=\dfrac{30}{100}[/tex]
The proportion of students have both a dog and a cat : [tex]P(C\cap D)=\dfrac{18}{100}[/tex]
Now, the conditional probability that a student who has a dog also has a cat will be :-
[tex]P(C|D)=\dfrac{P(C\cap D)}{P(D)}\\\\\\\Rightarrow\ P(C|D)=\dfrac{\dfrac{18}{100}}{\dfrac{45}{100}}\\\\\\\Rightarrow\ P(C|D)=\dfrac{18}{45}=\dfrac{2}{5}[/tex]
Hence, the probability that a student who has a dog also has a cat = [tex]\dfrac{2}{5}[/tex]
Answer:
3/5
Step-by-step explanation: I just did the test and got it right. This was after I tried 2/5 and got it wrong.
A right triangle whose base is 30 units is divided into two parts by a line drawn parallel to the base. It is given that the resulting right trapezoid has an area larger by 7,0 (which is sexagesimal) = 420 than the upper triangle, and that the difference between the height y of the upper triangle and the height z of the trapezoid is 20. if x is the length fo the upper base of the trapezoid, these statement lead to the relations 1/2z (x+30) = 1/2xy + 420, y - z = 20. The problem calls for finding the values of the unknown quantities x, y, and z. [Hint: By properties of similar triangles, y/(y+z) = x/30.]
Answer:
[tex]x=18[/tex]
[tex]y=60[/tex]
[tex]z=40[/tex]
Step-by-step explanation:
From the relations stablished in the problem we have the following equation system:
[tex]y-z=20[/tex] (equation 1)
[tex]\frac{1}{2} xy+420=\frac{1}{2}z(x+30)[/tex] (equation 2)
[tex]\frac{y}{y+z} =\frac{x}{30}[/tex] (equation 3)
From equation 1 we can find an expression of [tex]y[/tex] in terms of [tex]z[/tex] which we're going to call equation 4
[tex]y-z=20[/tex]
[tex]y=z+20[/tex] (equation 4)
We can then replace the equation 4 in the equation 2 in order to find an expression of [tex]x[/tex] in terms of [tex]z[/tex]
[tex]\frac{1}{2} xy+420=\frac{1}{2}z(x+30)[/tex]
[tex]\frac{1}{2} (xy+840)=\frac{1}{2}z(x+30)[/tex]
[tex]xy+840=z(x+30)[/tex]
[tex]x(z+20)+840=z(x+30)[/tex] (here we replaced the eq.4)
[tex]xz+20x+840=xz+30x[/tex]
[tex]xz+20x-xz=30z-840[/tex]
[tex]20x=30z-840[/tex]
[tex]10(2x)=10(3z-84)[/tex]
[tex]x=\frac{1}{2} (3z-84)[/tex] (equation 5)
Now, we can replace equations 4 & 5 inside the equation 3 so we can find the value of [tex]z[/tex]
[tex]\frac{y}{y+z} =\frac{x}{30}[/tex]
[tex]\frac{z+20}{z+20+z} =\frac{1}{30}*\frac{1}{2} (3z-84)[/tex]
[tex]\frac{z+20}{2z+20} =\frac{1}{30}*\frac{1}{2} (3z-84)[/tex]
[tex]\frac{z+20}{2(z+10)} =\frac{1}{2}*\frac{1}{30} (3z-84)[/tex]
[tex]\frac{1}{2}*\frac{z+20}{z+10} =\frac{1}{2}*\frac{1}{30} (3z-84)[/tex]
[tex]\frac{z+20}{z+10} =\frac{1}{10} (\frac{3z}{3}-\frac{84}{3})[/tex]
[tex]\frac{z+20}{z+10} =\frac{1}{10} (z-28)[/tex]
[tex]z+20 =\frac{1}{10} (z-28)*(z+10)[/tex]
[tex]10(z+20) =z^{2}+10z-28z-280[/tex]
[tex]10z+200 =z^{2}-18z-280[/tex]
[tex]z^{2}-28z-480=0[/tex]
This is a quadratic equation which has the form [tex]a*z^{2} +b*z+c=0[/tex]
where
[tex]a=1[/tex]
[tex]b=-28[/tex]
[tex]c=-480[/tex]
Then, we can find the solutions to this quadratic equation using the well-know quadatric formula which says that
[tex]z=\frac{-b}{2a}[/tex]±[tex]\frac{\sqrt{b^{2}-4ac} }{2a}[/tex]
then, replacing the values of a, b and c we find the values of z
[tex]z_{1}=\frac{-(-28)+\sqrt{(-28)^{2}-4(1)(-480)} }{2(1)}[/tex]
[tex]z_{1}=40[/tex]
[tex]z_{2}=\frac{-(-28)-\sqrt{(-28)^{2}-4(1)(-480)} }{2(1)}[/tex]
[tex]z_{2}=-12[/tex]
We have two possible values of z, but because we're trying to find the measure of trapezoid's height the result shouldn't be negative, so we keep only the positive value of z, then
[tex]z=40[/tex]
Now we may replace this value of z in the equations 4 & 5 in order to find the values of x & y.
[tex]y=z+20[/tex] (equation 4)
[tex]y=40+20[/tex]
[tex]y=60[/tex]
[tex]x=\frac{1}{2} (3z-84)[/tex] (equation 5)
[tex]x=\frac{1}{2} (3(40)-84)[/tex]
[tex]x=18[/tex]
So we've found the values of x, y, and z.
[tex]x=18[/tex]
[tex]y=60[/tex]
[tex]z=40[/tex]
PLEASE HELP ASAP!!! CORRECT ANSWERS ONLY PLEASE!!! THIS IS THE LAST DAY TO COMPLETE THIS ASSIGNMENT AND I DESPERATELY NEED TO FINISH THIS ASSIGNMENT WITH AN 100%.
Answer:
c. 7,999,999
Step-by-step explanation:
The number of possible phone numbers is the product of the number of possible digits in each position, less the excluded number:
8·10·10 · 10·10·10·10 - 1 = 8,000,000 -1 = 7,999,999
(a⁷ - a⁴) ÷ (a³ + a²)
Answer:
a^4 - a^3 + a^2 - 2a - (2)/(a + 1)
The simplified form of (a⁷ - a⁴) ÷ (a³ + a²) is a³ - a⁴.
The given expression is: (a⁷ - a⁴) ÷ (a³ + a²)
To simplify it:
Factor out common terms: a⁴(a³ - 1) / a²(a + 1)
Cancel out common factors: a⁴(a³ - 1) / a²(a + 1) = a³ - a⁴
Therefore, the simplified form of (a⁷ - a⁴) ÷ (a³ + a²) is a³ - a⁴.
Two teams are distributing information booklets. Team A distributes 60% more boxes of booklets than Team B, but each box of Team A’s has 60% fewer booklets than each box of Team B’s. Which of the following could be the total number of booklets distributed by the two groups?
A. 2,000
B. 3,200
C. 4,100
D. 4,800
E. 4,900
Answer:
C. 4,100
Step-by-step explanation:
"60% more" is represented by a multiplier of 1 + 0.60 = 1.60.
"60% fewer" is represented by a multiplier of 1 - 0.60 = 0.40.
__
Let b represent the number of booklets distributed by Team B. Then the number distributed by Team A is ...
1.60 × (0.40b) . . . . 60% more boxes, each with 60% fewer booklets
= 0.64b
Then the total distributed by both teams is ...
b + 0.64b = 1.64b = (164/100)b = (41/25)b
The only answer choice that is a multiple of 41 is ...
4,100 . . . choice C
__
For 4100 to be the number of booklets distributed by both teams, Team B will have distributed 2500 booklets, and Team A will have distributed 1600 booklets. Team A might have distributed 160 boxes of 100 booklets, while Team B might have distributed 100 boxes of 250 booklets.
Solve the Quadratics:
1) k^2-8k=0
2) a^2+5a=0
3) 6n^2+5n-25=0
4) 2x^2-11x-21=0
5) 2n^2+13n+19=4
Bonus Word Problem:
The larger leg of a right triangle is 7cm longer than its smaller leg. The hypotenuse is 8cm longer than the small leg. How many centimeters long is the smaller leg?
Answer:
Step-by-step explanation:
1 ) k²-8k = 0
k(k-8)=0
k = 0 or k=8
2) a²+5a=0
a(a+5) = 0
a=0 or a = - 5
3 ) 6n²+5n-25=0
delta = b²-4ac b =5 and a = 6 and c= - 25
delta = 5²-4(6)(-25) = 625 = 25²
n 1 = (-5+25)/12 = 20/12 = 5/3
n 2 = (-5-25)/12 = - 30/12 = -5/2
same method for 4) and 5)
1.) what is the domain of the following set? (-2,4), (0,3), (1,6), (-1,2)
a. (-2,-1,0,1)
b. (1,-2,1,6,3)
c. (-2,-1,0,1,2,3,6)
d. (4,3,6,-1,2)
e. (4,3,6,-1,-2)
Answer:
A. (-2,-1,0,1) .......
A regression model is used to forecast sales based on advertising dollars spent. The regression line is y=500+35x and the coefficient of determination is .90. Which is the best statement about this forecasting model?a. For every $35 spent on advertising, sales increase by $1.
b. Even if no money is spent on advertising, the company realizes $35 of sales.
c. The correlation between sales and advertising is positive.
d. The coefficient of correlation between sales and advertising is 0.81.
Answer:
The correlation between sales and advertising is positive.
Step-by-step explanation:
For every $35 spent on advertising, sales increase by $1
Is FALSE, since y = 500 + 35 x $35, sales increase more than $1
Even if no money is spent on advertising, the company realizes $35 of sales
Is FALSE, if no money is spent, the sales amount to $ 500 (when X = 0)
The coefficient of correlation between sales and advertising is 0.81
Is FALSE, since R² = 0.9. The coefficient of correlation = R = 0.94, not 0.81
HELP FASTTTTT PLEASE Assume that the following figures are drawn to scale. Use your understanding of congruence to explain why square ABCD and rhombus GHIJ are not congruent.
Answer:
see the explanation
Step-by-step explanation:
we know that
If two figures are congruent, then the corresponding sides and the corresponding angles are congruent
In this problem, the corresponding sides are congruent, but the corresponding angles are not congruent
therefore
The square ABCD and the rhombus GHIJ are not congruent
Two Geometrical Shape are Congruent, only when
1. Corresponding sides are equal
2. Corresponding Interior as well as Exterior Angles are equal.
3. Areas are equal.
⇒Square ABCD and Rhombus GHIJ, have length of their Corresponding side equal , but their interior angles are not equal.
So,⇒ Square ABCD NOT≅ to Rhombus GHIJ
This function f(x) has a domain of x = {-a, -b, a, b}.
In order, the x values are -a, -b, b, a.
In order, the f(x) values are 3c + 1, 2d - 5, 4d + 3, 6 - 2c.
Which values of c and d make this an even function?
a. c = -7 and d = 1/3
b. c = 5 and d = 1/3
c. c = -5 and d = -4
d. c = 1 and d = -4
e. c = -7 and d = -4
Answer:
d. c = 1 and d = -4
Step-by-step explanation:
If a function is even, then f(-x) = f(x). Graphically, this means it's symmetrical about the y-axis.
f(-a) = f(a)
3c + 1 = 6 − 2c
5c = 5
c = 1
f(-b) = f(b)
2d − 5 = 4d + 3
-2d = 8
d = -4
Therefore, c = 1 and d = -4.
An algebra tile configuration where the 2 largest tiles are labeled plus x squared, there are 4 tiles labeled negative x, where each tile is half the size of the largest tiles, and 6 tiles labeled minus that are each one-quarter the size of the largest tile. Which polynomial is represented by the algebra tiles? 2x2 – 4x – 6 2x2 + 4x + 6 –2x2 – 4x – 6 –2x2 + 4x + 6
The algebra tile configuration described represents the polynomial 2x^2 - 4x - 6.
Explanation:The algebra tile configuration you have described is representing the polynomial 2x2 - 4x - 6. This is deciphered as follows:
The 2 largest tiles labeled 'plus x squared' represent 2x2. For each of these tiles, the value is 'x squared' so two of them equals '2x squared'Four tiles half the size of the largest tiles are labeled 'negative x', thus representing -4x. If one large x tile represents 'x', then a half-size x tile represents '0.5x'. Since there are four such negative tiles, we get '4 * (-0.5x)' equals '-4x'.Six smallest tiles representing 'minus' account for the constant term of -6 since each of them is a quarter of the size of the largest tiles.In conclusion, the three parts together give us the polynomial: 2x2 - 4x - 6.
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The polynomial represented by the algebra tiles is [tex]\(2x^2 - 4x - 6\).[/tex]Therefore, the correct option is [tex]\(2x^2 - 4x - 6\).[/tex]
To determine which polynomial is represented by the given algebra tile configuration, we need to understand the values assigned to each type of tile.
Step 1:
Interpret the tiles.
The largest tiles represent [tex]\(x^2\),[/tex] the tiles labeled negative x represent [tex]\(-x\),[/tex] and the tiles labeled minus represent [tex]\(-1\).[/tex]
Step 2:
Assign values to the tiles.
Each largest tile represents [tex]\(x^2\),[/tex] so we have [tex]\(2x^2\).[/tex]
There are 4 tiles labeled negative x, each representing [tex]\(-x\),[/tex] so we have [tex]\(-4x\).[/tex]
There are 6 tiles labeled "minus," each representing [tex]\(-1\),[/tex] so we have [tex]\(-6\).[/tex]
Step 3:
Combine the terms.
Combining the terms, we get [tex]\(2x^2 - 4x - 6\).[/tex]
So, the polynomial represented by the algebra tiles is [tex]\(2x^2 - 4x - 6\).[/tex]Therefore, the correct option is[tex]\(2x^2 - 4x - 6\).[/tex]
The M&M jar has a square base with a length and width of 7 cm and a height of 6.5 cm. What would be the most reasonable lower limit for the number of M&M`s in the jar of the choices below?
A.10
B.100
C.1,000
D.10,000
Answer: I believe the answer would be 100 because there are or should be more than 100 M&M's in the jar.
Step-by-step explanation:
Show that the points A (-3, 2), B (-6, 4) and C (1, 8) are vertices of a right triangle.
Answer:
See below.
Step-by-step explanation:
For the triangle to be a right triangle there must be a pair adjacent sides which are at right angles to each other - that is whose slope product = -1.
Slope of AB = (4-2)/(-6- -3) = -2/3.
Slope of BC = (8-4)/ (1 - - 6) = 2/7
Slope of AC = (8-2) / (1 - -3) = 6/4 = 3/2.
Now 3/2 * -2/3 = -1 so sides AB and AC are at right angles and the 3 points are the vertices of a right triangle.
To confirm if the points A (-3, 2), B (-6, 4) and C (1, 8) are vertices of a right triangle, we use the Pythagorean theorem. After calculating the distances between each pair of points, we found that the square of the length of the longest side equals the sum of the squares of the lengths of the other two sides, proving that they form a right triangle.
Explanation:To show that the points A (-3, 2), B (-6, 4), and C (1, 8) are vertices of a right triangle, we need to check if the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides. This is known as the Pythagorean theorem. First, compute the distances between each pair of points using the distance formula:
AB = sqrt[(4-2)^2 + (-6-(-3))^2] = sqrt[2^2 + (-3)^2] = sqrt[4 + 9] = sqrt[13]
BC = sqrt[(8-4)^2 + (1-(-6))^2] = sqrt[4^2 + 7^2] = sqrt[16 + 49] = sqrt[65]
AC = sqrt[(8-2)^2 + (1-(-3))^2] = sqrt[6^2 + 4^2] = sqrt[36 + 16] = sqrt[52]
BC is the longest side, so we need to check if BC^2 = AB^2 + AC^2. Calculating, we find that 65 = 13 + 52, which is true. Therefore, points A, B, and C are vertices of a right triangle.
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The manager of a restaurant found that the cost to produce 300 cups of coffee is $30.43, while the cost to produce 500 cups is $49.83. Assume the cost C(x) is a linear function of x, the number of cups produced.
a) Find the formula for C(x)
b) What is the fixed (initial cost)
c) Find the total cost of producing 1200 cups
Answer:
a) C(x) = 1.33 + 0.097x
b) Fixed Initial cost = $1.33
c) C(1200) = $ 117.73
Step-by-step explanation:
a) Let's first define our x variable and y variable as:
x: Number of cups of coffee produced
y: Cost of producing
y is a function of x that in this problem is called C(x) so y = C(x).
No we are told that C(x) is a linear function. All linear functions follow the rule:
C(x) = mx+b
where m is the slope of the line and b is the intercept in the y - axis or the value of the function when x=0 . To find a formula for C(x) we can use the information given because these are two points of the line where
Point 1
x1= 300 and y1 = 30.43
Point 2
x2= 500 and y2 = 49.83
With these two points we can find the slope with the formula
m= y2-y1/x2-x1 = (49.83-30.43)/(500-300) = 19.42/200 = 0.097
so we have that;
C(x) = mx+b = 0.097x+b.
Now we have to know b the intercept in y.For this problem this is equivalent to the cost that we would have to pay if we did not produced any cup so b is our fixed initial cost. Because we have a point, we can replace it in the equation and solve for b. It doesnt matter which point we use.
C(x) = 0.097x + b
b = C(x) - 0.097x
With Point 2 = x = 500 and C(x) = 49.83
b = C(x) - 0.097x
b = 49.83 - (0.097 * 500) = 49.83 -48.5 = 1.33
So the final formula for C(x) is
C(x) = 0.097x + 1.33
b) As I said before, the initial cost or fixed cost is the cost incurred if we would not produce anything or mathematically when x = 0
C(x) = 0.097x + 1.33
C(0) = 0.097*0 + 1.33 = 0+1.33 = 1.33
The fixed cost is $ 1.33 that is the same as b parameter.
c) Now that we have an equation for C(x) we only need to replace for the point x = 1200
C(x) = 0.097x + 1.33
C(1200) = (0.097*1200) + 1.33 = 116.4 +1.33 = $ 117.73
The formula for the cost function C(x) is C(x) = $0.097x + $1.33, where $1.33 represents the fixed cost. Using this formula, the total cost of producing 1200 cups of coffee is $117.73.
Find the Cost Function C(x)
To find the cost function C(x), we need two points to determine a linear function: (300, $30.43) and (500, $49.83). First, find the slope (m) of the cost function using the formula m = (y2 - y1) / (x2 - x1), which in our case is m = ($49.83 - $30.43) / (500 - 300), so m = $19.40 / 200 = $0.097 per cup. The slope represents the variable cost per cup of coffee.
With the slope, we can use one of the points to find the y-intercept (b), the fixed or initial cost. Plug in the values into y = mx + b, so $30.43 = $0.097*300 + b, which gives us b = $30.43 - $29.10 = $1.33. Therefore, the formula for C(x) is C(x) = $0.097x + $1.33.
To find the total cost of producing 1200 cups, plug x = 1200 into the cost function: C(1200) = $0.097*1200 + $1.33, which calculates to C(1200) = $116.40 + $1.33 = $117.73.
Hence, the total cost of producing 1200 cups of coffee is $117.73
What do I fill in the boxes?
Answer:
see explanation
Step-by-step explanation:
Given
[tex]\frac{4x+1}{x^2-4x-12}[/tex] ← factorise the denominator
x² - 4x - 12 = (x - 6)(x + 2)
The fraction can now be expressed as
= [tex]\frac{4x+1}{(x-6)(x+2)}[/tex]
Split the numerator into its 2 parts, that is
= [tex]\frac{4x}{(x-6)(x+2)}[/tex] + [tex]\frac{1}{(x-6)(x+2)}[/tex]
On a coordinate plane, a curved line with minimum values of (negative 2, 0) and (1.05, negative 41), and a maximum value of (negative 0.5, 5), crosses the x-axis at (negative 2, 0), (0, 0), and (1.5, 0), and crosses the y-axis at (0, 0). Which statement is true about the end behavior of the graphed function? As the x-values go to positive infinity, the function’s values go to positive infinity. As the x-values go to zero, the function’s values go to positive infinity. As the x-values go to negative infinity, the function’s values are equal to zero. As the x-values go to negative infinity, the function’s values go to negative infinity.
Answer:
As the x-values go to positive infinity, the function’s values go to positive infinity.
Step-by-step explanation:
With the information given you can plot a rough graph (see attachment)
As the x-values go to positive infinity, the function’s values go to positive infinity. -> True
As the x-values go to zero, the function’s values go to positive infinity. -> False, x = 0 is between a maximum and a minimum
As the x-values go to negative infinity, the function’s values are equal to zero. -> False x-values go to negative infinity, the function's values go to positive infinite
As the x-values go to negative infinity, the function’s values go to negative infinity. False x-values go to negative infinity, the function's values go to positive infinite
Answer: As the x-values go to positive infinity, the function’s values go to positive infinity.
Step-by-step explanation:
just did this
Solve for y. −140=18+4(5y−2) Enter your answer in the box. y =
Answer:
y = - 7.5
Step-by-step explanation:
Given
- 140 = 18 + 4(5y - 2) ← distribute and simolify right side
- 140 = 18 + 20y - 8
- 140 = 10 + 20y ( subtract 10 from both sides )
- 150 = 20y ( divide both sides by 20 )
- 7.5 = y
a. Draw the image of ΔDEF after a rotation of 90° clockwise about the point (1,0). Label the image ΔD’E’F’.
b. Draw the image of ΔD’E’F’ after a reflection across the line x = 1. Label the image ΔD”E”F”.
Please help!!
Answer:
see the attachment
Step-by-step explanation:
(a) Clockwise rotation moves a point from (x, y) to (y, -x).
__
(b) Reflection across x=1 moves a point from (x, y) to (2-x, y).
Alisa says it is easier to compare the numbers in set a (45,000, 1,025,680) instead of set b (492,111, 409,867). 1.What is one way you could construct an argument justifying whether Alissa conjecture is true? 2. Is Alisa's conjecture true? Justify your answer. 3. Alisa wrote a comparison for Set B using ten thousand place. Explain what strategy she could have used.
Alisa's conjecture is true because the numbers in set A differ by two magnitudes, making the comparison easier. Alisa could have compared the leading digits in the ten thousand place to determine that the numbers in set B are relatively close in magnitude.
Explanation:In order to construct an argument justifying whether Alisa's conjecture is true, we can compare the magnitude of the numbers in each set. Looking at set A, we have 45,000 and 1,025,680. The first number has four digits, while the second number has six digits, indicating a difference of two magnitudes.
Now let's look at set B, which consists of 492,111 and 409,867. Both numbers in set B have six digits, so they are of the same magnitude.
Based on this analysis, we can conclude that Alisa's conjecture is true. It is indeed easier to compare the numbers in set A because their magnitudes differ by two, making the comparison more straightforward.
When comparing set B using the ten thousand place, Alisa could have used the strategy of looking at the leading digit in the ten thousand place. In this case, the leading digits are 4 and 4, indicating that the numbers in set B are relatively close in magnitude.
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Joseph says that in the number 9,999,999 all the digits have the same value. I'd Joseph correct? Explain Part B Describe the relationship between the value of the digits in the number
Answer:
Joseph is not correct
Each digit from the leftmost digit is ten times the digit before it
Step-by-step explanation:
* Lets explain how to solve the problem
- Any number formed from some digits, each digit has a place value
- Ex: 2,345 this number formed from 4 digits
The place value of 5 is ones ⇒ 5
The place value of 4 is tens ⇒ 40
The place value of 3 is hundreds ⇒ 300
The place value of 2 is thousands ⇒ 2,000
2,345 = 2,000 + 300 + 40 + 5
* Lets check our problem
∵ The number is 9,999,999
- The number formed from 7 digits, the digits have different places value
∴ They couldn't have the same value
∴ Joseph is not correct
∵ The number formed from 7 digits
- The place value of the leftmost digit is millions
- The place value of the digit before the million is hundred thousands
- The place value of the digit before the hundred thousands is
ten thousands
- The place value of the digit ten thousands is thousands
- The place value of the digit before thousands is hundreds
- The place value of the digit before hundreds is tens
- The place value of the digit before tens is ones
∴ 9,999,999 = 9,000,000 + 900,000 + 90,000 + 9,000 + 900
+ 90 + 9
∴ Each digit from the leftmost digit is ten times the digit before it