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In each triangle, M, N, and P are the midpoints of the sides. Name a segment parallel to the one given.

Answer:

g h hh h

Step-by-step explanation:

[tex](\stackrel{x_1}{-12}~,~\stackrel{y_1}{15})\qquad (\stackrel{x_2}{18}~,~\stackrel{y_2}{-13}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-13}-\stackrel{y1}{15}}}{\underset{\textit{\large run}} {\underset{x_2}{18}-\underset{x_1}{(-12)}}} \implies \cfrac{-28}{18 +12} \implies \cfrac{ -28 }{ 30 } \implies - \cfrac{14 }{ 15 }[/tex]

The code segment that assigns bonus correctly for all possible integer values of score is D, which uses nested if statements to implement the game's rules for assigning a value to bonus based on the value of score.

The code segment that assigns bonus correctly for all possible integer values of score is D:

IF(score < 50)

{

   bonus ← Ø

}

ELSE

{

   IF (score > 100)

   {

       bonus ← score (10)

   }

   ELSE

   {

       bonus ← score

   }

}

This code segment correctly implements the rules for assigning a value to bonus based on the value of score. It first checks if score is less than 50, and if so, it assigns 0 to bonus. If score is greater than or equal to 50, it checks if score is greater than 100, and if so, it assigns 10 times score to bonus. Otherwise, it assigns score to bonus. This covers all possible integer values of score and ensures that bonus is assigned correctly according to the game's rules.

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Complete question is in the image attached below

The values of x representing side length for which triangle is acute is given by 4, 5, and 6 .

For a triangle to be acute,

Sum of the squares of the two shorter sides must be greater than the square of the longest side.

Mathematically,

a^2 + b^2 > c^2

where a, b, and c are the side lengths of the triangle,

With c being the longest side.

The sides are given as 3, 6, and x.

Without loss of generality,

Assume that 3 and 6 are the shorter sides,

3^2 + 6^2 > x^2

Simplifying this inequality, we get,

45 > x^2

Taking the square root of both sides, we get,

6.71 > x

Since x must be an integer, the possible values of x are 4, 5, and 6.

Triangle inequality is satisfied,

Sum of any two sides of a triangle must be greater than the third side.

3 + 6 > x

3 + x > 6

6 + x > 3

Inequalities are all satisfied for x = 4, 5, and 6.

Therefore, there are 3 possible values of x is 4, 5, and 6 for which the triangle is acute.

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The distance from the base of the tower to the end of the wire is approximately 16.41 feet, and the height of the tower is approximately 23.41 feet.

What is Distance?

Distance is a numerical measurement of how far apart objects or points are. It is a scalar quantity, which means it only has a magnitude (size) and not a direction.

Let h be the height of the tower, and let d be the distance from the base of the tower to the end of the wire.

We are told that h = d + 7, and we know that the wire is 35 feet long. We can use the Pythagorean theorem to relate d and h to the length of the wire:

[tex]d^2 + h^2 = 35^2[/tex]

Substituting h = d + 7, we get:

[tex]d^2 + (d + 7)^2 = 35^2[/tex]

Expanding the left-hand side, we get:

[tex]2d^2 + 14d - 696 = 0[/tex]

Dividing both sides by 2, we get:

[tex]d^2 + 7d - 348 = 0[/tex]

We can solve this quadratic equation using the quadratic formula:

[tex]d =\frac{ (-b \pm \sqrt{(b^2 - 4ac)}) }{2a}[/tex]

where a = 1, b = 7, and c = -348. Substituting these values, we get:

d ≈ 16.41 or d ≈ -21.41

Since d represents a distance, it must be positive, so we take d ≈ 16.41.

Finally, we can use h = d + 7 to find the height of the tower:

h ≈ 23.41

Therefore, the distance from the base of the tower to the end of the wire is approximately 16.41 feet, and the height of the tower is approximately 23.41 feet.

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Answer:

g(y) = √(3/2 y)

Step-by-step explanation:

To find the inverse of a function, we need to solve for x in terms of y and interchange x and y. That is, we need to write the given function f(x) = 2/3x^2 in the form y = 2/3x^2 and then solve for x in terms of y.y = 2/3x^2

Multiplying both sides by 3/2, we get:

3/2 y = x^2

Taking the square root of both sides, we get:x = ± √(3/2 y)

Note that we have two possible values of x for each value of y, because the square root can be either positive or negative. However, for a function to have an inverse, it must pass the horizontal line test, which means that each value of y can only correspond to one value of x.Therefore, we need to restrict the domain of the original function to ensure that it is one-to-one. The simplest way to do this is to take the range of the function and use it as the domain of the inverse function.The range of f(x) = 2/3x^2 is all non-negative real numbers, or [0, ∞). Therefore, we can define the inverse function g(y) as:

g(y) = ± √(3/2 y)

where we choose the positive square root to ensure that the function is one-to-one.Thus, the inverse of the function f(x) = 2/3x^2 is:

g(y) = √(3/2 y)

with domain [0, ∞).

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Answer:

Angle AIC is vertical.

Step-by-step explanation:

Defn of vertical angles

Answer:

perimeter of plot=119m

circumference of pond=44m

volume of house=9000 m^3

surface area of room=248 m^2

Step-by-step explanation:

Answer:

1.Neither

2.Perpendicular

3.Perpendicular

4.Neither

5.Perpendicular

6.Perpendicular

7.Neither

8.Neither

9.Perpendicular

10.Neither

11. Perpendicular

12.Perpendicular

13.Neither

14.Neither

15.Neither

16.Neither

17.Parallel

18.Neither

here are the answers in order from top to bottom

The three positive numbers 72, 30, and 12 make up the total of 114.An object that can be counted, measured, or given a name is a number. The numbers, for instance, are 1, 2, 56, etc.

Suppose the three positive numbers are x, y and z.

If the summation of numbers is 114.

⇒ x + y + z = 114 --- (1)

⇒ z = 114 - x - y

Suppose the square sum of the number be S.

∴ S = x² + y² + z²

⇒ S = x²+ y² + (114 - x - y) ² --- (2)

To obtain the minimal value of S, we will optimize the function by partly differentiating it with respect to x and y and setting ds/dx = 0.

2x + 2(114- x - y) (-1) = 0

2x - 228 + 2x + 2y = 0

4x + 2y - 228 = 0

y = 114 - 2x --- (3)

Differentiate partially with respect to y as

2y + 2(114 - x - y) (-1) = 0

2y - 228 + 2x + 2y = 0

4y + 2x - 228 = 0

x = 228 - 2y --- (4)

Substitute the value of y = 114 - 2x in the equation (4),

x = 12 - 2(114 - 2x)

x = 12 - 228 + 4x

4x - x = 228 - 12

3x = 216

⇒ x = 72

By substituting x = 72 in equation (3)

y =2(72) - 114

y = 144 - 114

⇒ y = 30

Substitute the values of x and y in equation (1),

72 + (30) + z = 114

z = -72+114-30

z= 12

Thus, the three positive numbers whose sum is 114 will be 72, 30, and 12.

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Step-by-step explanation:

Please provide the figure then it will be easy to answer

The volume of the solid is given by the equation V = 36π (√2 - 1) using the triple integral.

A three-dimensional object's volume in three-dimensional space may be determined using the triple integral. The three-variable function is represented by the triple integral. If in space is a closed region, the region's entire volume may be expressed as V = Ddv, which is equivalent to V = D d x d y d z.

Cone: z = [tex]\sqrt{x^2+y^2}[/tex]

sphere: x² + y² + z² = 18

Here, we will use cylindrical coordinates to evaluate volume:

x = rcosθ , y = rsinθ, z = z

so, z = [tex]\sqrt{r^2cos^2\theta+r^2sin^2\theta} =r[/tex]

z = [tex]\sqrt{18-(x^2+y^2)} =\sqrt{18-r^2}[/tex]

r = [tex]\sqrt{18-r^2}[/tex]

r = 3

Finding limits,

[tex]Volume = \int\limits^2_0 \int\limits^3_0\int\limits^a_r {rdzdrd\theta} \, \\\\= \int\limits^2_0 \int\limits^3_0rz \ drd\theta\\\\= \int\limits^2_0 \int\limits^3_0 r(\sqrt{18-r^2}-r ) \ drd\theta\\\\[/tex]

Now, we have

[tex]\int\limits^3_0 {r\sqrt{18-r^2} } \, dr = -(9-18\sqrt{2} ) = 18\sqrt{2} -9[/tex]

Now the integral becomes,

Volume = 2π [(18√2-9) - 9]

= 2π x 18√2 - 18

V = 36π (√2 - 1)

Therefore, the volume of the solid is given by V = 36π (√2 - 1).

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Answer:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

invNorm(0.484)= 1.96

Therefore, the Z-score that has 48.4% of the distribution's area to its left is approximately 1.96.

To calculate the buoyant force that will be applied to a string of 2 inch OD 1.688 ID open end tubing, you will need to calculate the weight of the submerged portion of the tubing. This can be done using the following equation:

Weight (lbs) = (Height (ft) * Length (ft) * 0.052 lbs/ft3) * Fluid Density (ppg)

In your scenario, the Height is 10700 ft - 7000 ft = 3700 ft, the Length is 10700 ft, and the Fluid Density is 9.6 ppg.

Therefore, the Weight of the submerged portion of the tubing is:

Weight (lbs) = (3700 ft * 10700 ft * 0.052 lbs/ft3) * 9.6 ppg

Weight (lbs) = 10967 lbs

Now that you know the weight of the submerged portion of the tubing, you can calculate the Buoyant Force by subtracting the Weight of the submerged portion of the tubing from the Weight of the total Height of the tubing. This can be done using the following equation:

Buoyant Force (lbs) = (Height (ft) * Length (ft) * 0.052 lbs/ft3) * Fluid Density (ppg) - Weight (lbs)

Therefore, the Buoyant Force that will be applied to the string of tubing when it

The area of the outdoor carpet is 63 square foot.

What does a math area mean?

The amount of space occupied by a flat (2-D) surface or an object's shape is known as its area. An object's area is the space that its boundary encloses. The quantity of unit squares that completely encircle the surface of a closed figure is its area.

The dimensions of the outdoor carpet which is rectangle in shape are:

L=6 ft and w=12 ft.

The dimensions of the indoor carpet that is triangle in shape are:

b=3 ft and h=9 ft.

Now, area of the outdoor= Area of the green region-Area of triangular region

 Area = l * b - 1/2 * b * h

         = 6* 12 - 1/2 * 3 * 6

         = 72 - 9

         = 63 square foot

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The complete question is -

How many square feet of outdoor carpet will we need for this hole? 72 ft 9 ft 3 ft 6 ft 6 ft 12 ft

The range of tails that can appear when a coin is tossed ten times is represented by the random variable X, and its expected value is 5.

The probability of getting a tail on a single toss of a fair coin is1/2. since each toss is independent of the others, we're capable to model the number of tails in 10 tosses as a binomial random variable with parameters n = 10 and p = 1/2.

The anticipated value of a binomial random variable is presented through the expression

E( X) = n * p

Exchanging n = 10 and p = 1/2, we get

E( X) = 10 *1/2

E( X) = 5

Thus, the expected value of the random variable X, which represents the range of tails that come up while a coin is tossed 10 times, is 5.

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Use the vector law of addition to get the resulting vector is [tex](7i+2j-k)[/tex] and magnitude of resulting vector is [tex]3\sqrt{6}[/tex].

What is the resulting vector and its magnitude?

Vector addition can be defined as the sum of two or more vectors of corresponding components.

It is given that,

[tex]\vec{r}= < 4, -2 > \\\vec{s}= < 3,4,-1 >[/tex]

Given vectors can also be written as,

[tex]\vec{r}=4i-2j\\\vec{s}=3i+4j-k[/tex]

Add above vectors as follows:

[tex]\vec{r}+\vec{s}=(4i-2j)+(3i+4j-k)\\\\=(7i+2j-k)[/tex]

Therefore,

[tex]\vec{r}+\vec{s}= < 7,2,-1 >[/tex]

Show the resulting vector as follows:

Now calculate the magnitude of the vector [tex](\vec{r}+\vec{s})[/tex]

[tex]|\vec{r}+\vec{s}|=\sqrt{(7)^2+(2)^2+(-1)^2}\\=\sqrt{49+4+1}\\=\sqrt{54}\\=3\sqrt{6}[/tex]

Hence the resulting vector is [tex](7i+2j-k)[/tex] and magnitude of resulting vector is [tex]3\sqrt{6}[/tex].

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Answer:

2.

a) log base 10 of 100

b) The expression means that 10 to the 2nd power equals 100.

c) 2

3.

4. It would make sense that the value is between 1 and 2 because 10 to the 1st power is 10, and 10 to the 2nd power is 100. 50 is between 10 and 100 so the value would have to be between 1 and 2. This works because logs are the inverse function to exponentiation.

a) 2.6021

b) 3 and the value is exact because the base 10 in the log expression is technically the exponent when you convert it to exponent form. This works because logs are the inverse function to exponentiation. So 10 to the 3rd power would give you exactly 1000.

To conduct a hypothesis test comparing variances of independent samples from two populations, the test statistic will have an F-distribution.

The F-distribution is a probability distribution that describes the ratio of two independent chi-squared distributions divided by their degrees of freedom. In this case, the numerator and denominator degrees of freedom are based on the sample sizes and variances of the two populations being compared.

The null hypothesis for the F-test is that the variances of the two populations are equal, and the alternative hypothesis is that they are not equal. The F-test allows us to determine if the difference in variances is statistically significant, and if we reject the null hypothesis, we can conclude that the variances are significantly different.

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The equation of the line that is parallel to the given line and passes through the point (-3, 2) is 3x - 4y = -17.

If the line's equation is axe + by + c = 0 and the coordinates are (x1, y1).

To find the equation of a line parallel to a given line, we must first understand that parallel lines have the same slope. As a result, we must first determine the slope of the given line.

3x - 4y = -17 is the given line. To determine its slope, solve for y and write the equation in slope-intercept form:

-3x - 4y = -3x - 17 y = (3/4)x + (17/4)

This line has a 3/4 slope.

Now we want to find the equation of a parallel line that passes through the point (-3, 2). Because the new line is parallel to the given line, it has a slope of 3/4.

We can write the equation of the new line using the point-slope form of the equation of a line as:

y - y1 = m(x - x1)

where m represents the slope and (x1, y1) represents the given point (-3, 2).

When m = 3/4, x1 = -3, and y1 = 2, we get:

y - 2 = (3/4)(x - (-3))

y - 2 = (3/4)(x + 3)

Divide both sides by 4 to get rid of the fraction.

4y - 8 = 3x + 9

3x - 4y = -17

As a result, the equation of the parallel line that passes through the point (-3, 2) is 3x - 4y = -17.

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Answer:

FIRST ONE "Deb sold vases for two years, neither sold nor bought the next year and then sold bases for two more years"

Step-by-step explanation:

Notice the number of bases in debs collection is DECREASING as the years passes for the first and third period. This is she is selling her vases but in the middle the number is the same (two point in the same horizontal line) this means she neither sold nor bought any vase in that period.

Answer:

efrtrreeeeed

ggggffffff

Step-by-step explanation:

ggffffddeeeereeeer hhhhfdvjuhhhhhhhhjjjhhhhhhgfxd you and your family are doing well and your family are doing well and your dad said he had a good night with you don't have stich clock I WINNNNNN have stich clock out of it and your mom are the best ones that have stich you and Darshanie is a good night with my mom and my mom are the only ones who have a good weekend too and your dad said you and I WINNNNNN you and I will get it you and your family have a good day I love your family and your mom are doing well too and my mom I WINNNNNN and your dad said for you guys to come back to work tomorrow and your family are doing well and your family are doing well and your family are doing well and your family are

The mean weight of 16 fruit picked at random, then their mean weight will be greater than 414.72 grams 20% of the time

The mean value of a set of data, is the sum of the values in the data, divided by the number of data in the dataset.

The population mean = 408 grams

The population standard deviation = 32 grams

The mean of 16 fruits such that the probability of the mean is larger than the value is 0.2, can be obtained as follows;

The standard error of the mean = The standard deviation/√(Sample size)

Therefore;

The standard error = 32/√(16) = 8

The mean that corresponds to a probability of 0.2, which is the 80th percentile of the normal distribution, can be found using the z-score of the 80th percentile, which is about 0.84

Therefore, we get;

z = (Sample mean - 408)/8 = 0.84

The sample mean = 8 × 0.84 + 408 =  414.72

The sample mean = 414.72 grams

Therefore;

The mean weight of the 16 fruit such that their weight is greater than the mean weight of the population 20% of the time is 414.72 grams.

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Dale will receive $1200, Evelyn will receive $1800, and Frank will receive $1500 of the $4500 profit.

What is profit ?

Profit is a financial term that refers to the difference between the revenue earned by a business or individual and the expenses incurred in producing or delivering goods or services. In other words, profit is the amount of money that is left over after all the costs of doing business have been paid.

Profit is an important indicator of the financial health of a business or individual, as it measures how much money is being earned from a particular activity or investment. A positive profit indicates that revenues are higher than expenses, while a negative profit indicates that expenses are higher than revenues.

According to the question:
The total investment is $4000 + $6000 + $5000 = $15000.

The proportion of the profit that each person will receive is equal to their investment divided by the total investment:

Dale's proportion = $4000 / $15000 = 4/15

Evelyn's proportion = $6000 / $15000 = 2/5

Frank's proportion = $5000 / $15000 = 1/3

To find each person's share of the profit, we multiply the total profit by their proportion:

Dale's share = (4/15) x $4500 = $1200

Evelyn's share = (2/5) x $4500 = $1800

Frank's share = (1/3) x $4500 = $1500

Therefore, Dale will receive $1200, Evelyn will receive $1800, and Frank will receive $1500 of the $4500 profit.

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The value of the triple integral is 7/180.

To evaluate the triple integral, we first need to set up the limits of integration. Since the solid tetrahedron T is defined by the four vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1), we can use these points to set up the limits of integration as follows:

For z, we integrate from the bottom of the solid (z = 0) to the top (z = 1 - x - y).

For y, we integrate from the left side of the solid (y = 0) to the right (y = 1 - x).

For x, we integrate from the back of the solid (x = 0) to the front (x = 1).

Therefore, the triple integral can be written as:

∫∫∫ t 7x^2 dv = ∫∫∫t 7x^2 dV

= ∫[0,1] ∫[0,1-x] ∫[0,1-x-y] 7x^2 dz dy dx

∫[0,1] ∫[0,1-x] ∫[0,1-x-y] 7x^2 dz dy dx

= ∫[0,1] ∫[0,1-x] 7x^2 (1 - x - y) dy dx

= ∫[0,1] [7x^2 (1 - x) (1/2 - x/3)] dx

= 7/180

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The answer of the given question is (a) TR = p(40 - (4/3)p), or TR = 40p - (4/3)p² , (b)  TC = 9q + (3/10)q² + 30 , (c) π = 40p - (4/3)p² - 9q - (3/10)q² - 30 , (d) TR ≈ 342.67 , (e) the profit when 20 units are produced is approximately RM188.27 , (f)  the profit when 7 units are produced is approximately -RM24.44, indicating a loss , (g) the output required to obtain a profit of RM100 is approximately 8.78 units.

An equation is  mathematical statement that asserts yhe equality of two expressions. It typically consists of variables, constants, and mathematical operations like addition, subtraction, multiplication, and division, among others. Equations are often used to solve problems, to model real-world phenomena, and to describe mathematical relationships.

(a) The total revenue is given by TR = p x q. Substituting the demand equation q = 40 - (4/3)p, we get TR = p(40 - (4/3)p), or TR = 40p - (4/3)p².

(b) The total cost is given by TC = q x AC. Substituting the given average cost function, we get TC = 9q + (3/10)q² + 30.

(c) The profit is given by π = TR - TC. Substituting the expressions we found in parts (a) and (b), we get π = 40p - (4/3)p² - 9q - (3/10)q² - 30.

(d) At the break even point, the firm earns zero profit, so we set π = 0 and solve for q. Substituting the expression we found in part (a) for p, we get:

0 = 40p - (4/3)p² - 9q - (3/10)q² - 30

0 = 40(40/3 - (3/4)q) - (4/3)(40/3 - (3/4)q)² - 9q - (3/10)q² - 30

0 = 533.33 - 51.25q - 0.22q^2

Solving for q using the quadratic formula, we get:

q = (51.25 ± sqrt(51.25² - 4(-0.22)(533.33))) / 2(-0.22)

q ≈ 22.75 or q ≈ 206.58

We reject the solution q ≈ 206.58 because it is outside the relevant range of output, which is between 0 and 40. Therefore, the total output at the break even point is approximately 22.75 units. To find the total revenue at the break even point, we substitute q = 22.75 into the demand equation from part (a) and get:

p = (40/3) - (3/4)q

p ≈ 15.08

TR = p x q

TR ≈ 342.67

(e) To find the profit when 20 units are produced, we substitute q = 20 into the expression for profit we found in part (c) and get:

π = 40p - (4/3)p² - 9q - (3/10)q² - 30

π ≈ 188.27

Therefore, the profit when 20 units are produced is approximately RM188.27.

(f) To find the profit when 7 units are produced, we substitute q = 7 into the expression for profit we found in part (c) and get:

π = 40p - (4/3)p² - 9q - (3/10)q² - 30

π ≈ -24.44

Therefore, the profit when 7 units are produced is approximately -RM24.44, indicating a loss.

(g) To find the output required to obtain a profit of RM100, we set the profit equation equal to 100 and solve for q:

Profit = TR - TC

100 = pq - ACq

100 = (40-(4/3)p)*q - (9+(3/10)q+30/q)*q

100 = (40-(4/3)p - 9q - 3q²/10)

Multiplying by 10 and rearranging terms, we get a quadratic equation in q:

3q² + 91q - 310 = 0

Solving for q using the quadratic formula, we get:

q = (-91 ± sqrt(91² - 43(-310)))/(2*3)

q ≈ 8.78 or q ≈ -29.44

Since the quantity produced cannot be negative, the output required to obtain a profit of RM100 is approximately 8.78 units.

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Answer:

Genevieve should use a three digit code instead of a two digit code for her locker combination because a three digit code has more possible combinations than a two digit code, which makes it more secure.

With a two digit code, there are only 100 possible combinations (10 possible digits for the first number and 10 possible digits for the second number). On the other hand, with a three digit code, there are 1,000 possible combinations (10 possible digits for each of the three numbers). This means that it would take a potential thief 10 times longer to guess the correct combination for a three digit code compared to a two digit code, assuming they are using a brute force method of trying all possible combinations.

Therefore, using a three digit code increases the security of the locker and decreases the likelihood of someone gaining unauthorized access.

Answer:

more security.

Step-by-step explanation:

If you have a 2 digit code people will find it. but with 3, it makes it harder to find so people can't open her locker.

So the solution equation is 6x + 3 and the total miles is equal to 6x + 3.

Where x represents the length of Lincoln Park School.

A linear equation is an algebraic equation of the form y=mx+b. m is the slope and b is the y-intercept. The above is sometimes called a "linear equation in two variables" where y and x are variables. 

Mile is a unit of measurement that equals 1760 yards or approximately 1.6 kilometer's. It the mostly used in the continent of North America.

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Answer:

  (a, b)  -2

  (c) -1, -1.5, -2, -2.5

Step-by-step explanation:

You want the average and instantaneous rates of change at various times in year 3, given the money in the bank after t years is 180+3t-t².

The amount at the beginning of year 3 is ...

  180 +3t -t² = 180 +t(3 -t)

  180 +2(3 -2) = 182 . . . . . . t=2

The amount at the end of year 3 is ...

  180 +3(3 -3) = 180 . . . . . . t=3

The amount increased by 180 -182 = -2 thousand dollars.

This change occurred in one year, so the average rate of change is ...

  change/years = -2/1 = -2 thousand dollars per year

The derivative of the amount function will give its instantaneous rate of change:

  da/dt = 3 -2t

The values of t at the beginning of the quarters in year 3 are ...

  t = 2: da/dt = 3 -2·2 = -1 thousand per year at start of 1st quarter

  t = 2.25: da/dt = 3 -2(2.25) = -1.5 thousand per year at start of Q2

  t = 2.50: da/dt = 3 -2(2.50) = -2 thousand per year at start of Q3

  t = 2.75: da/dt = 3 -2(2.75) = -2.5 thousand per year at start of Q4