Answer: 8 runners
Step-by-step explanation:
If you mean that " if each person runs one more mile than the previous runner" then:
First person = 1
Second person = 1+1
.....
(1)+(1+1)+(1+1+1)+(1+1+1+1)+(1+1+1+1+1)+(1+1+1+1+1+1)+(1+1+1+1+1+1+1)+(1+1+1)
1 + 2 + 3 + 4 + 5 + 6 + 7 + 3 =31
(the last runner runs less miles then needed)
8 runnersIf alpha and beta are the roots of 3x²-4x+1. Find
a) alpha²-beta²
b) 2/alpha²+2/beta²
c)2/alpha²-2/beta²
From Vieta's formulas, we have
[tex]3x^2 - 4x + 1 = 3 (x - \alpha) (x - \beta) \\\\ ~~~~~~~~~~~~~~~~~ = 3x^2 - 3 (\alpha + \beta) + 3\alpha\beta \\\\ \implies \begin{cases} \boxed{\alpha + \beta = \frac43} \\\\ \boxed{\alpha\beta = \frac13} \end{cases}[/tex]
a) Note that
[tex](\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2 \\\\ \implies \alpha^2 - \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta - 2\beta^2[/tex]
We have exact values for [tex]\alpha+\beta[/tex] and [tex]2\alpha\beta[/tex], but sadly not for [tex]-2\beta^2[/tex]. This means we'll need to solve for [tex]\beta[/tex] explicitly.
By factorizing, we have
[tex]3x^2 - 4x + 1 = (3x - 1) (x - 1) = 0 \\\\ \implies x = \dfrac13 \text{ or } x = 1[/tex]
Then the value of [tex]\alpha^2-\beta^2[/tex] depends on which is chosen to be the larger root, and
[tex]\alpha^2 - \beta^2 = \dfrac1{3^2} - 1^2 = \boxed{-\dfrac89} \text{ or } \alpha^2 - \beta^2 = 1^2 - \dfrac1{3^2} = \boxed{\dfrac89}[/tex]
While we're at it, we can also immediately find
[tex]\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\dfrac43\right)^2 - 2\cdot\dfrac13 = \dfrac{10}9[/tex]
b) Combine fractions to get
[tex]\dfrac2{\alpha^2} + \dfrac2{\beta^2} = \dfrac{2\beta^2 + 2\alpha^2}{\alpha^2\beta^2} \\\\ ~~~~~~~~~~~~~ = 2 \cdot \dfrac{\alpha^2 + \beta^2}{(\alpha\beta)^2} \\\\ ~~~~~~~~~~~~~ = 2\cdot \dfrac{\frac{10}9}{\left(\frac13\right)^2} \\\\ ~~~~~~~~~~~~~ = \boxed{20}[/tex]
c) Combine fractions again.
[tex]\dfrac2{\alpha^2} - \dfrac2{\beta^2} = 2 \cdot \dfrac{\beta^2 - \alpha^2}{(\alpha\beta)^2} \\\\ ~~~~~~~~~~~~~ = -2\cdot \dfrac{\pm\frac89}{\left(\frac13\right)^2} \\\\ ~~~~~~~~~~~~~ = \boxed{\pm16}[/tex]
The base of a solid right pyramid is a square with an edge length of n units. The height of the pyramid is n − 1 units.
The correct option is C. One-third n²(n − 1) units³ = ⅓ n²(n-1) units³; volume of the pyramid.
What is referred as the solid right pyramid?A right pyramid is one that has the upper end straight just above centroid of a base. A regular pyramid is thus a subset of a right pyramid.
As, per the given question;
Edge Length = n units
Height of the pyramid = n - 1 units
The formula for volume of a solid right pyramid is;
Volume = ⅓Ah
Where, A is the area of the base.
h is the height of the pyramid
Since, pyramid has the square base;
Calculate the area;
Area, A = edge × edge
A = n × n
A = n²
Put the value of area in volume;
Volume = ⅓Ah
Substitute n² at A and n - 1 at h.
Solve the expression;
Volume = ⅓ × n² × (n - 1)
Volume = ⅓n²(n-1).
Therefore, the volume of the solid right pyramid is calculated as V = ⅓n²(n-1) units³.
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The complete question is -
A solid right pyramid has a square base with an edge length of n units. The height of the pyramid is n minus 1 units.
Which expression represents the volume of the pyramid?
A. One-third n(n − 1) units³
B. One-third n²(n − 1)² units³
C. One-third n²(n − 1) units³
D. One-third n³(n − 1) units³
Hi i need help with my work
The equation to express the relationship will be y = 5.05 + 2.45x.
How to illustrate the information?Worth of the card = $5.06
It should be noted that the card increases by $2.45.
Therefore, the equation to express the relationship will be:
y = 5.06 + (2.45 × x)
where x = number of years.
y = value of the card.
y = 5.05 + 2.45x
In conclusion, the equation to express the relationship will be y = 5.05 + 2.45x.
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If m/CDE is a straight angle, DE bisects m/GDH, mZGDE = (8x-1), m/EDH = (6x +
m/CDF = 43°, find each measure.
x =
m/GDH=
mZFDH=
m/FDE =
Answer:
From the way the problem is worded:
CDE is a straight angle made up of three angles: CDF of measure 43, FDG of unknown measure and GDE of measure 8x + 1.
EDH is outside CDE and is of measure 6x + 15.
Since DE bisects GDH to form GDE and EDH, those latter two angles are equal and thus 8x + 1 = 6x + 15 and x = 7 and substituting 7 for x, both angles are equal to 57 degrees.
Since CDF is 43 and GDE is 57 and CDE is 180 in total, then FDG = 180 - 43 - 57 = 80 degrees.
Find the coordinates of the missing endpoint if M is the midpoint of Ab B(-7 , 16) and M (-2 , 8)
If M is the midpoint of AB , B(-7,16) and M(-2,8) then coordinates of the missing endpoint A are (3,0).
As given in the question,
M is the midpoint of AB
Let coordinates of A(x₁ ,y₁) , B(x₂ ,y₂) and M(x ,y)
So,
x = (x₁ + x₂)/2
⇒-2 = (x₁ + (-7))/2
⇒ x₁ = 7-4
⇒x₁ =3
y = (y₁+ y₂)/2
⇒8 = (y₁ +16)/2
⇒ y₁ = 16-16
⇒ y₁ =0
Therefore, if M is the midpoint of AB , B(-7,16) and M(-2,8) then coordinates of the missing endpoint A are (3,0).
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The sumerian mathematical system was based on units of sixty, ten, and six and survives in what modern system?
The sumerian mathematical system was based on units of sixty, ten, and six and survives in what modern system time measurement.
What is sumerian mathematical system?
Sumer (a region of Mesopotamia, present-day Iraq) is sometimes referred to as the Cradle of Civilization because it was the origin of writing, the wheel, agriculture, the arch, the plow, irrigation, and many other accomplishments.
We really know more about ancient Sumerian and Babylonian mathematics than we do about early Egyptian mathematics because the Sumerians invented the first known writing system, the pictographic cuneiform script, which uses wedge-shaped symbols engraved on baked clay tablets. In fact, we even have math and geometry problems that seem like school exercises.
The base number 60 was created by adding a "celestial" 6 and a "common" 10 in the Sumerian "sexagesimal" system.
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The options are
A. (6, 3)
B. (1, -4)
C. (2, 1)
D. (0, 3)
Answer:
Correct answer is D.
Step-by-step explanation:
A. 6 = 3-3 ×
B. 1 = -4-3 ×
C. 2 = 1-3 ×
D. 0 = 3-3 ✓
Multiply.
(−5 5/8)⋅(−2 1/3)
9/2 I think
I hope that helped!
Answer:
13.125 or 105/8 or 13 1/8
Step-by-step explanation:
What is the most specific name for the figure? I NEED THE ANSWER
The most specific name for the figure in the image is a Rectangle. Option B
This is further explained below.
What is the most specific name for the figure?Generally, According to the principles of Euclidean geometry, a Rectangle is considered to be a regular quadrilateral since it has four sides of equal length amongst opp two sides
Another way to describe it is as a rectangle with two neighboring sides that have the same length.
The term coordinate is pronounced exactly the same as o-ordinate. When used as an adjective, these phrases might signify to have the same status or rank, to have something to do with coordination, or to have something to do with an intersection of indices.
In conclusion, A Rectangle is the most accurate description of the shape that can be seen in the picture. Alternative B
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Which table represents a linear function?
Answer:
Option C.
Step-by-step explanation:
For a table to be linear, the table must have a constant and even rate of change, meaning as X, the input, increases by 1, Y, the output, increases evenly by a random number we're gonna name "a". Now, a, in this case, is -8. The numbers increase evenly by -8(a), which would classify that function as linear. Thus, option C. is the correct answer.
One packet of hot cocoa weighs 1.38 ounces. How much will 8 packets weigh?
Answer:
Your answer will be 11.04
Step-by-step explanation:
1.38 × 8= 11.04
4. The store offers a similar deal if you purchase the $139.95 workouts but with a 20% off discount
instead of 15% for purchases over $1000. The home gym she'd really like, the Shredmaster 5000,
costs $1199 before-tax. Write a new function, g(x) for this second deal and calculate the
before-tax cost of the Shredmaster 5000.
the price of the Shredmaster 5000 will be $ 959.2 and the function is g (x) = 0.8 x for x > 1000.
If the purchase is made for $ 1000 and above. the discount = 20 %
Now the cost of Shredmaster 5000 = $ 1199
Now, the price after the discount will be
Price = 1199 - 20 % of 1199
Price = 1199 - 20 / 100 × 1199
Price = 1199 - 0.20 × 1199
Price = 1199 - 239.8
Price = $ 959.2
The function will be:
g (x) = x - 0.2 x
g (x) = 0.8 x for x > 1000.
Therefore, the price of the Shredmaster 5000 will be $ 959.2 and the function is g (x) = 0.8 x for x > 1000.
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for 1983 through 1989, the per capita consumption of chicken in the u.s. increased at a rate that was approximately linear. in 1983, the per capita consumption was 33.3 pounds, and in 1989 it was 48 pounds. write a linear model for per capita consumption of chicken in the u.s. let t
In 1995, we would anticipate a per capita consumption of 54.3 pounds of chicken.
What is per capita consumption?Per capita consumption is the yearly use of services and goods by each individual, calculated by dividing the total population's use of goods and services. This variable measures personal economic well-being directly.So,
According to the data, per capita consumption was 39.7 pounds in 1983 and 47 pounds in 1989.Per capita consumption is the yearly use of services and goods by each individual, calculated by dividing the total population's use of goods and services. This variable measures personal economic well-being directly.Assume t=0 corresponds to the year 1980.As a result, we can express the given data as ordered pairs (3,39.7) and (9,47).We can compute a linear function that passes through these points.Let us first calculate the slope of the linear function:
[tex]m=\frac{y_2-y_1}{t_2-t_1}=\frac{47-39.7}{9-3}=\frac{7.3}{6}=\frac{73}{60}[/tex]The required linear function can be written as:
[tex]y-39.7=\frac{73}{60}(t-3)[/tex]When we simplify this, we get:
[tex]y=\frac{73}{60} t+\frac{721}{20}[/tex]To calculate per capita consumption in 1995, enter t=15 into this function.
[tex]y=\frac{73}{60}(15)+\frac{721}{20}=54.3[/tex]Therefore, in 1995, we would anticipate a per capita consumption of 54.3 pounds of chicken.
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The correct question is given below:
For 1983 through 1989, the per capita consumption of chicken in the u.S. Increased at a rate that was approximately linear. In 1983, the per capita consumption was 39.7 pounds, and in 1989 it was 47 pounds. Write a linear model for per capita consumption of chicken in the u.S. Let t represent time in years, where t = 3 represents 1983. Let y represent chicken consumption in pounds. What would you expect the per capita consumption of chicken to be in 1995? What would you expect the per capita consumption of chicken to be in 1995?
lol last one before i have to watch another lesson love yall
Hey there!
4a + 1 + a + 3
= 4a + 1 + 1a + 3
COMBINE the LIKE TERMS:
= (4a + 1a) + (1 + 3)
= (4a + a) + (1 + 3)
= 4a + a + 1 + 3
= 5a + 4
Therefore, your answer should be:
5a + 4
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
Which of the following is equal to 7 1/3
A. √7
B. √7.3
C. 73
D. 3√7
a board measures 212 inches. how many feet is this? [ ? ] ft. give your rounded answer with one decimal place.
Answer:17.67
Step-by-step explanation:
question one-half cup of black beans provides 15% of the potassium you need daily. you must get the remaining 2890 milligrams from other sources. how many milligrams of potassium should you consume daily? question one-half cup of black beans provides 15% of the potassium you need daily. you must get the remaining 2890 milligrams from other sources. how many milligrams of potassium should you consume daily?
The amount of potassium you should consume daily is 510 miligrams if 15 percent of potassium you daily need.
Let the total amount of sources of nutrient you consumed daily be x miligrams.
According to the given question.
One-half cup of black beans provides 15 percent of the potassium.
Also, you get remaning nutrients from 2890 miligrams of other sources.
⇒ you get 85 percent of remaining nutrients from other sources.
So, we can say that
85 percent of x = 2890 miligrams
⇒ 85/100 × x = 2890 miligrams
⇒ x = (2890 × 100)/85
⇒ x = 3400
So, the total amount of source of nutrients is 3400.
Therefore,
The amount of potassium you should consume daily
= 15 percent of 3400
= (15/100) × 3400
= 15 × 34
= 510 miligrams
The amount of potassium you should consume daily is 510 miligrams if 15 percent of potassium you daily need.
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Reduce the following notes and determine error of closure bs hi fs elevation remarks 2.25 1149.92 bm start 3.58 5.99 12.56 4.66 5.68 10.66 4.55 9.88 10.15 2.33 1.02 1.66 4.59 bm start
The error in the closure is equal to the difference in the actual value of evaluation and the measured value of evaluation of the same point which is equal to 0.02 units.
What do you mean by the term error of closure?There is a discrepancy between the existing coordinates and the computed coordinates. The amount by which a quantity obtained by a series of related measurements differs from the true or fixed value of the same quantity sometimes known as a closure mistake. The difference between the real value of a series of angles and its exact value in theory. likewise known as an angular mistake of closure. The degree to which two azimuth readings for a line that were obtained through distinct surveys or along different routes do not perfectly match one another. is also known as an azimuth closure error.
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The temperature on Wednesday at 5:00 p.m. was 15°F. In the following 12 hours, the temperature dropped 22°F. The temperature on Thursday at 5:00 p.m. was 27°F.
Part A: What was the temperature on Thursday at 5:00 a.m.? Explain or show how you got your answer.
The temperature on Thursday at 5:00 a.m. was
The temperature required on Thursday at 5:00 a.m. was observed as -7°F
given,
The temperature on Wednesday at 5:00 p.m. = 15°F
the temperature dropped 22°F in the following 12hours
The temperature on Thursday at 5:00 p.m. was 27°F.
the temperature on Thursday at 5:00 a.m. is calculated with a vertical number line.
number line - A number line is defined as the number marked on the line calibrated into an equal number of units.
For example -1, 0, 1, and so on.
Thus, the temperature on Thursday at 5:00 a.m. is
=15° - 22°
= -7°
The temperature required on Thursday at 5:00 a.m. was observed as -7°F
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Find the distance between (0, -8) and (3, -2). Express your answer as a simplified radical, if possible.
O 5√3
O22.5
O √45
O 3√5
Answer:
3√5
Step-by-step explanation:
You use the Distance Formula to solve this.
d=√(x2-x1)^2 + (y2-y1)^2
Plug in your points.
d=√(3-0)^2 + (-2- -8)^2
Subtract and square.
d=√9 + 36
Add
d=√45
The square root of 45 is 6.70, so you would look for radicals.
3√5 is 45. <3
As the sample size increases, the margin of error _____. a. decreases b. increases c. stays the same d. becomes negative
As the sample size increases, the margin of error a. decreases
What is a margin of error?In a random survey sample, the margin of error is a statistical measure that takes into account the discrepancy between actual and anticipated findings. Simply said, you may determine the degree of unpredictability in data and research results using the margin of error.
The number of distinct samples that are measured or observations that are used in a survey or experiment is known as the sample size. For instance, your sample size is 100 if you examine 100 samples of soil for signs of acid rain. If 30,500 completed questionnaires from an online survey were returned, your sample size is 30,500.
Hence, as the sample size increases, the margin of error decreases.
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HELP PROONTOOOOOOOOOOOOOOOOOOOOOOOOOO
Answer:
x - 5 = 65
Step-by-step explanation:
The shape of a dome can be
modeled by the equation h =
- 2d^2 + 100 where h is the height
(in feet) of the dome from the
floor d feet from its center. How
far from the center of the dome
is the height 50 feet?
Answer:
5 feet
Step-by-step explanation:
We are told that the height of the dome can be modelled by the equation:
[tex]\boxed{h = -2d^2 + 100}[/tex]
The question is essentially asking us: for what value of d will the value of h be 50 feet?
To solve this, we have to substitute h in the equation with 50 and then solve for d.
∴ [tex]50 = -2d^2 + 100[/tex]
⇒ [tex]50 + 2d^2 + 100[/tex]
⇒ [tex]2d^2 = 100 - 50[/tex]
⇒ [tex]d^2 = \frac{100 - 50}{2}[/tex]
⇒ [tex]d^2 = 25[/tex]
⇒ [tex]d = \sqrt{25}[/tex]
⇒ [tex]d = \bf 5[/tex]
This means that 5 feet from the center of the dome, the height of the dome is 50 feet.
does anybody know this? i’ve been struggling for hours
12 lies between 9 and 16 so √12 lies between 3 and 4.
Perfect square-: Those whole numbers in which we can pair the factors are known as perfect square.
We can write the number 40 as 2 × 2 × 2 × 5. This is not a perfect square because we cannot pair the factors.
Factor, in mathematics, a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder. For example, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly.
We write 40−−√ to be the square root of 40, and can investigate between which two whole numbers it lies.
Consider the table of perfect squares below:
Whole number 1 2 3 4 5 6 7 8 9 10 11 12
Whole number squared 1 4 9 16 25 36 49 64 81 100 121 144
The number 40 lies between 36 and 49 hence 40−−√ lies between 6 and 7.
Similarly, 12 lies between 9 and 16 so √12 lies between 3 and 4.
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A house cost $150,000 and you give 10%. The amount of mortgage is. a. 135,000
B.130,000 C.125,000 D.145, 000
Answer:
Your answer is C.125,000
Step-by-step explanation:
The cost of a school dance is given by the formula: c = 2p + 500 where c represents the total cost in dollars, and p the number of students attended. How many students can attend the school dance if we have $1,672 to spend on the dance?
Answer:
586 students can attend the school dance if we have $1,672 to spend on the dance.
Step-by-step explanation:
1672 = 2p + 500 Subtract 500 from both sides of the equation
1172 = 2p Divide both sides of the equation by 2
586 = p
PLEASE HELP!!!! I WILL MARK BRAINLIEST!!!
Find a quadratic equation that models the decline in physical recorded music revenue (not digital).Let x=number of years since 2000 and y be the physical recorded music revenue in millions of dollars.
PLEASE HELP!!!! I WILL MARK BRAINLIEST!!!
The rate of decrease in the revenue reduces each year, resulting in an
approximate quadratic equation model that is concave upwards.
The variables are;
x = Number of years since 2,000
y = Revenue for physically recorded music in millions of dollars
Find out -
To model the decline using a quadratic equation.
The general form of the quadratic equation is; y = a·x² + b·x + c
When y = 5547, x = 8
We get;
5547 = a·8² + b·8 + c = 64·a + 8·b + c
5547 = 64·a + 8·b + c...(1)
When y = 3113, x = 11
We get;
3113 = a·11² + b·11 + c = 121·a + 11·b + c
3113 = 121·a + 11·b + c...(2)
When y = 2056, x = 14
We get;
2056 = a·14² + b·14 + c = 196·a + 14·b + c
2056 = 196·a + 14·b + c...(3)
Subtracting equation (2) from equation (1) gives;
5547 - 3113 = 64·a + 8·b + c - (121·a + 11·b + c) = -57·a - 3·b
2434 = -57·a - 3·b...(4)
Subtracting equation (3) from equation (2) gives;
3113 - 2056 = 121·a + 11·b + c - (196·a + 14·b + c) = -75·a - 3·b
1057 = -75·a - 3·b...(5)
Subtracting equation (5) from equation (4) gives;
2434 - 1057 = -57·a - 3·b - (-75·a - 3·b) = 18·a
1377 = 18·a
a = 76.5
From, equation (4), 2434 = -57·a - 3·b, we have;
2434 = -57 × 76.5 - 3·b
Therefore;
b = -2264.8
From equation (1), we get;
5547 = 64 × 76.5 + 8×(-2264.8) + c
Therefore;
c = 5547 - (64 × 76.5 + 8×(-2264.8)) = 18769.
c = 18769.
Therefore, the quadratic model is;
Verifying, we have in year 2012, x = 12
y = 76.5×12² - 2264.8×12 + 18769. = 2607.74
Thus ,the completed table and comparison of the predicted physical revenue and actual values is presented in the attachment;
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does x+6=x have one solution, no solution, or infinite solutions?
Answer:
infinite solution
Step-by-step explanation:
because thats the right answers
Is the following statement true or false?
3 is not a character of that set. answer is False
Think About the Process The length of cell A is 2x107 m. The length of cell B is 0.00000001 m. What is the ratio of cell A's length to cell B's length? Use pencil and paper. Is it easier to find the
ratio when the numbers are expressed in scientific notation or in standard form? Explain your reasoning
The ratio of cell A's length to cell B's length is [tex]2\times10^{15} :1[/tex]
In mathematics, a ratio is a comparison of two or more numbers, indicating their magnitude relative to each other. A ratio compares two quantities by division. The dividend or divisor is called the antecedent and the divisor or divisor is called the sequence.Scientific notation is a form of representing very large or very small numbers in a simpler form.It is given that length of cell A is [tex]2\times10^7m[/tex] which is in scientific notation and length of cell B is 0.00000001 m which is in standard form.
On converting length of B in scientific notation , we get
[tex]0.00000001 =1.0\times10^{-8}m[/tex]
The ratio of length A to length B is given by
[tex]\frac{2\times10^7}{1.0\times10^{-8}} =\frac{2\times10^{15}}{1}[/tex]
which can be written as [tex]2\times10^{15} :1[/tex]
Yes, finding ratios is easier when numbers are represented in scientific notation. This is because scientific notation is a mathematical methodology used to describe very large or very small numbers in a compact and simple form, raised to powers of ten.
Hence , it is easier to calculate or solve mathematical operations using numbers in scientific notation.
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