Cars existed before 1950. estimate the fuel efficiency of a 1940-model-year car. explain how you got your answer.

Answer:

2.50 p + 1.50p

Step-by-step explanation:

if you buy an item x amount of times you would times the two to figure out the total cost of that type of item then add with the other

The final integral is:∫₀^² ∫₀^²π ρ⁹ cosθsinθ dθdρ= 49/80 [sin(2π/3) - sin(4π/3)] [2⁹ - 1⁹]≈ 1.24. Therefore, the required answer is 1.24.

The given integral is:

∫∫∫ _E xyz dv where E lies between the spheres rho = 1 and rho = 2 and above the cone ϕ = π/3.

To evaluate the given integral, we use cylindrical coordinates.

We know that the cylindrical coordinates are (ρ,θ,z).

Using cylindrical coordinates, we have:x = ρcosθy = ρsinθz = z

Thus, the given integral becomes ∫∫∫ _E ρ³cosθsinθz dρdθdz

We know that the region E lies between the spheres ρ = 1 and ρ = 2 and above the cone ϕ = π/3.

The equation of the cone is ϕ = π/3.

We convert this to cylindrical coordinates by using z = ρcosϕ and ϕ = tan⁻¹⁡(z/ρ)sin(π/3) = √3/2tan⁻¹⁡(z/ρ)

Thus, the cone is given by the inequality tan⁻¹⁡(z/ρ) ≥ √3/2ρ ≥ 1The boundaries for the remaining variables are θ = 0 to 2π and ρ = 1 to 2.

Thus, the integral becomes:

∫₀^² ∫₀^²π ∫_(√3ρ/2)^(2ρ) ρ⁵cosθsinθz dzdθdρ

Evaluating the integral we get:

∫₀^² ∫₀^²π [z²ρ⁵cosθsinθ/2]_(√3ρ/2)^(2ρ) dθdρ= ∫₀^² ∫₀^²π 7ρ⁹/4 cosθsinθ dθdρ= 7/4 ∫₀^² ∫₀^²π ρ⁹ cosθsinθ dθdρ

We can easily evaluate the integral above using integration by parts.

We have to use integration by parts twice.

The final integral is:∫₀^² ∫₀^²π ρ⁹ cosθsinθ dθdρ= 49/80 [sin(2π/3) - sin(4π/3)] [2⁹ - 1⁹]≈ 1.24.

Therefore, the required answer is 1.24.

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A scattergram with a general trend of the points from the lower left to the upper right indicates that the Pearson r value was positive.10. A. −0.78. The absolute value of Pearson's correlation  ranges from 0 to 1, with 0 indicating no correlation, and 1 indicating perfect correlation.

An r value of −0.78 is closer to -1 than an r value of −0.68, indicating that it has a stronger correlation.11. B. positive correlation. When two variables have a positive correlation, it means that as one variable increases, so does the other.12. B. relatively longer toes. A Pearson r of +0.84 indicates a positive correlation between the length of a person's toes and the number of pairs of shoes they own.

So, on average, people who own relatively more pairs of shoes have relatively longer toes.13. C. −1;+1. The Pearson r correlation coefficient is a value that ranges from -1 to 1, with -1 indicating a perfect negative correlation, 0 indicating no correlation, and 1 indicating a perfect positive correlation.14. C. Linear regression. The amount of variation in scores on a Sociology test that can be predicted by the number of hours students studied can be calculated using linear regression.

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The value of the union of sets A and B is, P(A ∪ B) is 0.64.

The union of two sets means the total elements in both the sets combined.

Given: sets P(A) = 0.38, P(B) = 0.83, and  intersection P(A ∩ B) = 0.57

We need to find the union of P(A ∪ B).

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Let us substitute the given values in the formula.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

=0.38+0.83-0.57

=1.21-0.57

=0.64

Therefore, the value of union of sets P(A ∪ B) is 0.64.

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Answer:

9/14.

Step-by-step explanation:

The probability is given as a fraction or percentage.

9:5 odds in favour of a horse winning  means the horse will win 9 races for every 5 it does not win.

So the probability that this horse will win is 9/(9+5)

= 9/14.

Answer:

|x-10 y-4|

Step-by-step explanation:

Let the initial position where the football is thrown up the air be x and y.

After the ball is thrown up the air it moves 10 feet west 4 feet south because of the gust of wind.

Therefore the change in x is by 10 feet west and is at the point x - 10 and the change in y is by 4 feet south and is at the point y - 4.

Thus the matrix representing the position of the football is |x-10 y-4|.

Please mark as brainliest if answer is right

Have a great day, be safe and healthy  

Thank u  

XD

C R E D I T:  mohit890

Regarding the risk-return graph for weights when ry = -0.80, 0.00, 0.60, the correct graph would depend on the specific data and context provided. It is not possible to determine the correct graph without additional information.

To calculate the expected returns and expected standard deviations of a two-stock portfolio with a correlation coefficient of 0.80, we can use the following formulas:
Expected return of a two-stock portfolio:
E(Rp) = w1 * E(R1) + w2 * E(R2)
Expected standard deviation of a two-stock portfolio:
E(σp) = √(w1^2 * E(σ1)^2 + w2^2 * E(σ2)^2 + 2 * w1 * w2 * ρ * E(σ1) * E(σ2))
where:
w1 = weight of stock 1
w2 = weight of stock 2
E(R1) = expected return of stock 1
E(R2) = expected return of stock 2
E(σ1) = expected standard deviation of stock 1
E(σ2) = expected standard deviation of stock 2
ρ = correlation coefficient

a) w1 = 1.00
Expected return of a two-stock portfolio:
E(Rp) = 1.00 * 0.13 + 0.00 * 0.17 = 0.13
Expected standard deviation of a two-stock portfolio:
E(σp) = √(1.00^2 * 0.04^2 + 0.00^2 * 0.05^2 + 2 * 1.00 * 0.00 * 0.80 * 0.04 * 0.05) = 0.04

b) w1 = 0.85
Expected return of a two-stock portfolio:
E(Rp) = 0.85 * 0.13 + 0.15 * 0.17 = 0.1355
Expected standard deviation of a two-stock portfolio:
E(σp) = √(0.85^2 * 0.04^2 + 0.15^2 * 0.05^2 + 2 * 0.85 * 0.15 * 0.80 * 0.04 * 0.05) = 0.0422

c) w1 = 0.55
Expected return of a two-stock portfolio:
E(Rp) = 0.55 * 0.13 + 0.45 * 0.17 = 0.145
Expected standard deviation of a two-stock portfolio:
E(σp) = √(0.55^2 * 0.04^2 + 0.45^2 * 0.05^2 + 2 * 0.55 * 0.45 * 0.80 * 0.04 * 0.05) = 0.0481

d) w1 = 0.25
Expected return of a two-stock portfolio:
E(Rp) = 0.25 * 0.13 + 0.75 * 0.17 = 0.1575
Expected standard deviation of a two-stock portfolio:
E(σp) = √(0.25^2 * 0.04^2 + 0.75^2 * 0.05^2 + 2 * 0.25 * 0.75 * 0.80 * 0.04 * 0.05) = 0.0546

e) w1 = 0.05
Expected return of a two-stock portfolio:
E(Rp) = 0.05 * 0.13 + 0.95 * 0.17 = 0.167
Expected standard deviation of a two-stock portfolio:
E(σp) = √(0.05^2 * 0.04^2 + 0.95^2 * 0.05^2 + 2 * 0.05 * 0.95 * 0.80 * 0.04 * 0.05) = 0.0566

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According to the given dot plot there are 18 students in the class.

3. From the given dot plot, we can see there are 18 students in the class.

4. In the class there are 18 students, survey is done on what kind of music each student like and what kind of Music CDs each student brought.

Therefore, according to the given dot plot there are 18 students in the class.

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The statements that are true when there exists a constant rate of change between x and y are:

1. The graph of the relationship between x and y is a straight line.
2. The slope of the line represents the constant rate of change between x and y.
3. The value of the slope is the same for any two points on the line.
4. The equation that represents the relationship between x and y can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
5. The value of the y-intercept is the y-coordinate of the point where the line crosses the y-axis.

Let's break down each statement:

1. The graph of the relationship between x and y is a straight line:
When there is a constant rate of change between x and y, the graph will be a straight line. This means that the points representing the relationship between x and y will lie on a straight line when plotted on a graph.

2. The slope of the line represents the constant rate of change between x and y:
The slope of a line is a measure of how steep or flat the line is. In this case, when there is a constant rate of change between x and y, the slope of the line will be the same for any two points on the line. It represents the rate at which y changes for every unit change in x.

3. The value of the slope is the same for any two points on the line:
As mentioned earlier, the slope represents the constant rate of change between x and y. Regardless of which two points we choose on the line, the ratio of the change in y to the change in x will always be the same.

4. The equation that represents the relationship between x and y can be written in the form y = mx + b, where m is the slope and b is the y-intercept:
When there is a constant rate of change between x and y, we can express their relationship using an equation in the form y = mx + b. The slope, represented by the variable m, will be the coefficient of x, and the y-intercept, represented by the variable b, will be the value of y when x is equal to 0.

5. The value of the y-intercept is the y-coordinate of the point where the line crosses the y-axis:
The y-intercept is the point where the line representing the relationship between x and y crosses the y-axis. It represents the value of y when x is equal to 0.

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Answer:

No it is irrational number.

Step-by-step explanation:

because 2pie cannot be expressed as a quotient of interger.

Answer:

\( l(\widehat {KM}) = 15.07 \: units\)

Step-by-step explanation:

\( Central\: angle \:(\theta) = 96\degree \)

Radius (r) = 9 units

\(l(\widehat {KM}) = \frac{ \theta}{360 \degree} \times 2\pi r \\ \\ l(\widehat {KM}) = \frac{96 \degree}{360 \degree} \times 2 \times 3.14 \times 9 \\ \\ l(\widehat {KM}) = \frac{96 }{360} \times 2 \times 3.14 \times 9 \\ \\ l(\widehat {KM}) = \frac{5,425.92}{360}\\ \\ l(\widehat {KM}) = 15.072\\ \\ l(\widehat {KM}) = 15.07 \: units\)

Answer:

subtrahend

Step-by-step explanation:

The number which is subtracted from another number is called "subtrahend".

10 - 2 = 8

In this subtraction, the first number, "10" is called the minuend.

The second number, "2" is called the subtrahend.

The third number "8" is the difference.

Answer:

7m³ - 11m² + 5m + 17

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Distributive Property

Algebra I

Step-by-step explanation:

Step 1: Define

3m³ - 4m² + 8 - (-4m³ + 7m² - 5m - 9)

Step 2: Simplify

The unit tangent vector to the space curve at time t = 2 is therefore \(\(\vec{T} = \frac{1}{\sqrt{21}}\vec{i} - \frac{4}{\sqrt{21}}\vec{j} + \frac{2}{\sqrt{21}}\vec{k}\)\)

What is a Unit tangent vector?

The direction of a curve at a particular point is depicted by the unit tangent vector, which is a vector. The direction the curve is traveling in at that location is shown by a vector of length 1. The unit tangent vector is frequently represented by the letters\(\(\vec{T}\) or \(\hat{T}\)\)

Using the given vector function, we can get the unit tangent vector to the space curve at the point (t = 2) by doing the following steps:

1. To get the velocity vector, calculate the derivative of the vector function.

2. Calculate the velocity vector at time t = 2 to determine the tangent vector.

The unit tangent vector is produced by normalizing the tangent vector.

The derivative of the vector function \(\(\vec{r}(t) = t\vec{i} - t^2\vec{j} + (2t-1)\vec{k}\)\) is found as follows:

\(\(\vec{v}(t) = \vec{r}'(t) = \frac{d\vec{r}}{dt} = \vec{i} - 2t\vec{j} + 2\vec{k}\)\)

We may calculate the velocity vector at time t by using the formula: \(\(\vec{v}(2) = \vec{i} - 2(2)\vec{j} + 2\vec{k} = \vec{i} - 4\vec{j} + 2\vec{k}\)\)

The curve at (t = 2) is represented by the tangent vector in this vector.

The tangent vector is normalized as follows to produce the unit tangent \(\(\vec{T} = \frac{\vec{v}(2)}{|\vec{v}(2)|}\)\)

Using the Euclidean norm, determine the size of \(\(\vec{v}(2)\)\):

\(\(|\vec{v}(2)| = \sqrt{\vec{v}(2) \cdot \vec{v}(2)} = \sqrt{1^2 + (-4)^2 + 2^2} = \sqrt{1 + 16 + 4} = \sqrt{21}\)\)

The unit tangent vector is thus:\(\(\vec{T} = \frac{1}{\sqrt{21}}\vec{i} - \frac{4}{\sqrt{21}}\vec{j} + \frac{2}{\sqrt{21}}\vec{k}\)\)

The unit tangent vector to the space curve at time t = 2 is therefore \(\(\vec{T} = \frac{1}{\sqrt{21}}\vec{i} - \frac{4}{\sqrt{21}}\vec{j} + \frac{2}{\sqrt{21}}\vec{k}\)\)

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Answer:

\(\frac{27y^6}{8x^{12}}\)

Step-by-step explanation:

1) Use Product Rule: \(x^ax^b=x^{a+b}\).

\((\frac{3x^{-5+2}{y^3}}{2z^0yx}) ^3\)

2) Use Negative Power Rule: \(x^{-a}=\frac{1}{x^a}\).

\((\frac{3\times\frac{1}{x^3} y^3}{2x^0yx} )^3\)

3) Use Rule of Zero: \(x^0=1\).

\((\frac{\frac{3y^3}{x^3} }{2\times1\times yx} )^3\)

4) use Product Rule: \(x^ax^b=x^{a+b}\).

\((\frac{3y^3}{2x^{3+1}y} )^3\)

5) Use Quotient Rule: \(\frac{x^a}{x^b} =x^{a-b}\).

\((\frac{3y^{3-1}x^{-4}}{2} )^3\)

6) Use Negative Power Rule: \(x^{-a}=\frac{1}{x^a}\).

\((\frac{3y^2\times\frac{1}{x^4} }{2} )^3\)

7) Use Division Distributive Property: \((\frac{x}{y} )^a=\frac{x^a}{y^a}\).

\(\frac{(3y^2)^3}{2x^4}\)

8) Use Multiplication Distributive Property:  \((xy)^a=x^ay^a\).

\(\frac{(3^3(y^2)^3}{(2x^4)^3}\)

9) Use Power Rule: \((x^a)^b=x^{ab}\).

\(\frac{27y^6}{(2x^4)^3}\)

10)  Use Multiplication Distributive Property:  \((xy)^a=x^ay^a\).

\(\frac{26y^6}{(2^3)(x^4)^3}\)

11) Use Power Rule: \((x^a)^b=x^{ab}\).

\(\frac{27y^6}{8x^12}\)

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Answer:

\(\displaystyle \frac{27y^{6}}{8x^{12}}\)

Step-by-step explanation:

\(\displaystyle \biggr(\frac{3x^{-5}y^3x^2}{2z^0yx}\biggr)^3\\\\=\biggr(\frac{3x^{-5}y^2x}{2}\biggr)^3\\\\=\frac{(3x^{-5}y^2x)^3}{2^3}\\\\=\frac{3^3x^{-5*3}y^{2*3}x^3}{8}\\\\=\frac{27x^{-15}y^{6}x^3}{8}\\\\=\frac{27y^{6}x^3}{8x^{15}}\\\\=\frac{27y^{6}}{8x^{12}}\)

Notes:

1) Make sure when raising a variable with an exponent to an exponent that the exponents get multiplied

2) Variables with negative exponents in the numerator become positive and go in the denominator (like with \(x^{-15}\))

3) When raising a fraction to an exponent, it applies to BOTH the numerator and denominator

Hope this helped!

Answer:

38

Step-by-step explanation:

50 - 12 is 38, so since she had 12 dollars left to spend she had wasted 38 dollars.

The correct significant figure of 0.0879/0.98 is 0.08989796. (rounded to six significant figures)

The meaningful digits in a measured or computed number are known as significant figures. They are used to convey the degree of uncertainty in a value and represent the accuracy of the measurement. You must first establish the number of significant figures in each value being utilized before you can execute a computation with significant figures. When doing the computation, use the same number of decimal places as the value with the fewest significant figures. The final step is to round the result to the appropriate number of significant digits.

According to the question

Both numbers in the calculation 0.0879/0.98 have four significant digits. The result, after performing the calculation, is 0.08989796. We look at the final digit (7) in the response to determine the right amount of significant digits to round to. The preceding digit (9) is rounded up if the digit is 5 or above. In this instance, the solution, rounded to six significant numbers, is 0.0899.

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Lina hops linearly and Kavier hops exponentially

The equations of the functions are Lina: y = 2x + 3 and Kavier: y = 3(2)^x

Linear pattern

The table of values represent the given parameter

For Lina's distance from the tree, we can see that

As the hop increases by 1, the distance covered increases constantly by 2 feet

This represents a linear function

Exponential pattern

For Kavier's distance from the tree, we can see that

As the hop increases by 1, the distance covered doubles

This represents an exponential function

Lina Linear Function

From the question, we have the following parameters that can be used in our computation:

(0, 3) and (1, 5)

A linear function is represented as

y = mx + c

Where

c = y when x = 0

So, we have

y = mx + 3

Using the points, we have

m + 3 = 5

m = 2

So, we have

y = 2x + 3

Kavier Exponential Function

Here, we have

(0, 3) and (1, 6)

An exponential function is represented as

y = ab^x

Where

a = y when x = 0

So, we have

y = 3b^x

Using the points, we have

3b = 6

b = 2

So, we have

y = 3(2)^x

Hence, Kavier's equation is y = 3(2)^x

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To factorise the quadratic expression x² + 6x + 7, we need to find two binomials whose product equals the original expression.

One way to do this is to use the fact that (a + b)² = a² + 2ab + b², which can be rearranged to give:

a² + 2ab + b² = (a + b)²

Using this identity, we can rewrite the expression x² + 6x + 7 as:

x² + 6x + 7 = x² + 2(3)(x) + 3² - 3² + 7

Notice that we added and subtracted 3² = 9 inside the parentheses. Now we can use the identity above to write:

x² + 6x + 7 = (x + 3)² - 2² + 7

Simplifying the expression inside the parentheses gives:

x² + 6x + 7 = (x + 3)² - 4

Therefore, we have factored the quadratic expression x² + 6x + 7 as:

x² + 6x + 7 = (x + 3)² - 4

Answer:

Step-by-step explanation:

if you can use negative numbers then   you could use 21 and -15

so

adding them together is  21 + (-15) = 6

and then subtracting  21 - (-15) =

                                     21 + 15 = 36

do you think that would work ?  :/

Answer:

21 &-15

Step-by-step explanation:

Plz mark brainly =)

Answer:

\(3\pi\)

Step-by-step explanation:

The circumference of a circle is \(2\pi r\).

If we want to find the circumference of this semi-circle, we can find the circumference if it was a whole circle then divide by 2.

\(2 \cdot \pi \cdot r\\2 \cdot \pi \cdot 3\\6 \cdot \pi\\ 6\pi\)

Now we know the circumference of the whole circle.

To find the circumference of half the circle we divide by 2.

\(6\pi \div 2 = 3\pi\)

Hope this helped!

a. The significance level of the test is the probability of rejecting the null hypothesis 1 - F(k). b. The value of k at the midpoint of this interval is the solution. c. the exact relationship between the number of chips and the temperature depends on the details of the chip fabrication process and the testing procedure.

(a) Suppose the rejection set of the test is {k or more chips fail in one day}. To find the significance level of the test as a function of the number of chips tested, we need to calculate the probability of rejecting the null hypothesis when it is true, which is the probability of observing k or more failures in one day when the chips are operating normally. This probability is given by the cumulative distribution function (CDF) of the exponential distribution with mean years, evaluated at the time period of one day:

P(reject null | null is true) = P(k or more failures in one day | normal operation) = 1 - F(k)

where F(k) is the CDF of the exponential distribution with mean years, evaluated at k/365. The significance level of the test is the probability of rejecting the null hypothesis when it is true, so we have:

significance level = P(reject null | null is true) = 1 - F(k)

(b) To find the number of chips that must be tested at temperature degree Celsius so that the significance level is , we need to solve the equation:

1 - F(k) =for k, where F(k) is the CDF of the exponential distribution with mean years, evaluated at k/365 and is the desired significance level. Since the exponential distribution is a continuous distribution, we can use a numerical method to solve this equation. One way is to use a root-finding algorithm such as the bisection method or Newton's method. For example, using the bisection method, we can start with an initial interval [a, b] that brackets the solution and iteratively bisect the interval until we obtain an interval of desired width that contains the solution. The value of k at the midpoint of this interval is the solution.

(c) If we raise the temperature during the test, the number of chips we need to test will decrease. This is because the higher temperature increases the failure rate of the chips, making it easier to detect contamination problems. However, the exact relationship between the number of chips and the temperature depends on the details of the chip fabrication process and the testing procedure, so we cannot give a general answer to this question without more information.

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Answer:

(a) Amy can buy 5 shirts and 4

(b) The combination is correct because it satisfies both inequalities

(c) The two possible ways to show that she can buy is 5 shirt and 4 pants are;

(1) Mathematically

(2) Graphically

Step-by-step explanation:

The given parameters are

The cost of each shirt = $40

The cost of each pant = $50

The amount Amy plans to spend ≤ $400

Let x represent the number of shirts Amy buys and let y represent the number of pants she buys, therefore, we have;

40 × x + 50 × y  ≤ 400...(1)

x + y ≥ 5...(2)

Making y the subject of both inequalities gives;

For the inequality (1), we have;

40 × x + 50 × y  ≤ 400

y ≤ (400/50) - (40/50) × x = 8 - (4/5)·x

y ≤ 8 - (4/5)·x

For the inequality (2), we have;

x + y ≥ 5

y ≥ 5 - x

Plotting both inequalities using the chart function in Microsoft Excel gives;

(a) As seen from the graph of the system of inequalities, Amy can buy 5 shirts and 4

(b) The combination of 5 shirts and 4 pants as a solution is correct because the value of the total number of the combination is larger than 5 (5 + 4 = 9 > 5) and the cost of 5 shirts plus 4 pants = $400

(c) The two possible ways to show that the combination of shirts and pants that Amy can buy is 5 shirt and 4 pants is a possible solution are;

(1) Mathematically, x = 5, y = 4

Therefore, y ≤ 8 - (4/5) × x gives;

4 ≤ 8 - (4/5) × 5 which is correct and

From y ≥ 5 - x gives;

4 ≥ 5 - 5 = 0, Which is also correct

(2) By plotting a graph of the system of inequalities as included

Answer: Yes that is correct Explanation:

Answer:

-7/4

Step-by-step explanation:

When you find the perpendicular slope, you take the opposite reciprocal of the original slope. For 4/7, the opposite is -4/7, and the reciprocal of that is -7/4.

The original amount is 12000

Fraction to daughter = 1/2

Fraction to son = 1/2 × 1/2 = 1/4

Brother = 1/6

The remainder is 1000. The fraction for the remainder will be:

= 1 - (1/2 + 1/4 + 1/6)

= 1 - (6/12 + 3/12 + 2/12)

= 1 - 11/12

= 1/12

Therefore, let the original amount be x

1/12x = 1000

x = 12 × 1000

x = 12000

The original amount is 12000.

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\( {m}^{2} + 5m + 7 \\ \\ put \: m = - 2 \\ \\ = {( - 2)}^{2} + 5 \times ( - 2) + 7 \\ = 4 - 10 + 7 \\ = 11 - 10 \\ = 1\)

\(\huge \boxed{\sf 1}\\\\\\\sf Evaluate\ for\ m=-2\\\\(-2)^2+5(-2)+7\\\\4+(-10)+7\\\\=1\)

Answer:

Woody arrived  

27 -  9 =  18  minutes before Barbie did.

Step-by-step explanation:

When you are working with Distance=speed-time problems remember that if you know two of the values you can calculate the third one,

Barbie: She walks at a speed of  70  metres per minute.  

Woody: Cycles  3  times as fast:   →3 × 70 = 210 m/min

He cycled for  9  minutes.

210 m/min is his speed.

9  minutes is the time he took.

Distance  =  speed  ×  time

D  =  210  × 9   =  1890  metres,

Barbie: walked the same distance of  1890  metres

Her speed was  70  m/min

The time she took:  

Time= \(\frac{Distance}{Speed}\)

T i m e  =   \(\frac{1890}{70}\) =  27  minutes.

Woody arrived  27  −  9  =  18  minutes before Barbie did.

In a four-candidate race with 107 votes cast, the smallest number of votes a winning candidate can have is 28.

In a four-candidate race decided by plurality, the winning candidate is the one who receives the most votes, regardless of whether that number of votes constitutes a majority (more than 50%) of the total votes cast.

To determine the smallest number of votes a winning candidate can have in a four-candidate race with 107 votes cast, we can assume that the other three candidates each receive an equal number of votes, say x. Then, the winning candidate must receive more votes than each of the other three candidates.

So, the minimum number of votes the winning candidate can receive is x + 1.

The total number of votes cast in the election is:

x + x + x + (x + 1) = 4x + 1

Since we know that 4x + 1 = 107, we can solve for x:

4x + 1 = 107

4x = 106

x = 26.5

Since x must be a whole number, we can round up to x = 27.

Then, the minimum number of votes the winning candidate can have is:

27 + 1 = 28

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