A can is to be made to hold a litre of oil. Find the radius of the can that will minimize the cost of the metal to make the can. (1L = 1000 cm)

Unfortunately, we don't have enough information to determine the area of the garden. We would need to know the dimensions of the backyard and/or the garden to calculate their areas.

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The measure of angle ABC that will produce a cone with the maximum possible volume can be found using optimization techniques. Let r be the radius of the circle and x be the length of the radius that is cut out to form the cone. Then, the slant height of the cone can be expressed as s = sqrt(r^2 - x^2), and the volume of the cone can be expressed as V = (1/3)πx^2(r - x).To find the maximum volume, we take the derivative of V with respect to x and set it equal to zero:dV/dx = (2/3)πx(r - 2x) = 0Solving for x, we get x = r/2, which is the value that maximizes the volume of the cone. Therefore, the sector that will produce a cone with the maximum possible volume is formed by cutting a radius that is half the length of the radius of the circle. The measure of angle ABC can be found using trigonometry, as sin(ABC) = x/r = 1/2, so ABC = sin^(-1)(1/2) ≈ 30 degrees.

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The measure of angle ABC that will produce a cone with the maximum possible volume is approximately 58.49 degrees.

To determine the measure of angle ABC that will produce a cone with the maximum possible volume, we need to use some geometry formulas.

First, we need to understand that the volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.

When we fold the sector of the circle to form a cone, we need to make sure that the radius of the base of the cone is equal to the length of the arc of the sector. Let's call this length x.

Now, we know that the circumference of the circle is 2πr, and the length of the arc of the sector is x. Therefore, the measure of the angle of the sector is (x/2πr) * 360 degrees.

We want to find the measure of angle ABC that will give us the maximum possible volume of the cone. To do this, we need to maximize the value of r and h.

Using some trigonometry, we can see that sin(ABC/2) = (x/2r). Rearranging this formula, we get r = x/(2sin(ABC/2)).

Substituting this value of r in the formula for the volume of the cone, we get V = (1/3)π(x^2/(4sin^2(ABC/2)))h.

To maximize this volume, we need to maximize both x and h. We know that x is fixed, so we need to maximize h.

Using some more trigonometry, we can see that h = rcos(ABC/2) = (x/2) * cot(ABC/2).

Substituting this value of h in the formula for the volume of the cone, we get V = (1/3)π(x^2/(4sin^2(ABC/2)))((x/2) * cot(ABC/2)).

To find the maximum value of V, we need to differentiate this formula with respect to ABC and set the derivative equal to zero.

After some calculations, we get tan(ABC/2) = 2/3. Solving for ABC, we get ABC = 2tan^-1(2/3) ≈ 58.49 degrees.

Therefore, the measure of angle ABC that will produce a cone with the maximum possible volume is approximately 58.49 degrees.

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The commission of the salesperson is $750 if he received a commission of 25% of the difference between the auction price and the dealership price.

The salesperson's commission can be calculated by first finding the dealership price, which is 30% more than the auction price of $10,000.

30% of $10,000 = $3,000

Dealership price = $10,000 + $3,000 = $13,000

Next, we need to find the difference between the dealership price and the auction price

$13,000 - $10,000 = $3,000

The salesperson's commission is 25% of this difference

25% of $3,000 = $750

Therefore, the salesperson's commission is $750.

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According to formed equations, the price of one teddy bear and one doll is $7 and $20 respectively.

Let us represent the teddy bear as x and dolls as y. Now, framing the equation for Dolly and Gwinnett.

Cost of teddy bear × number + cost of dolls × number = total expenditure

14x + 3y = 158 : equation 1

8x + 12y = 296 : equation 2

Divide the equation by 4

2x + 3y = 74 : equation 3

Subtraction equation 3 from equation 1

14x + 3y = 158

- 2x + 3y = 74

12x = 84

x = 84/12

x = $7

So, the cost of teddy bear = $7

keep the value of x in equation 3 to find the cost of y

2×7 + 3y = 74

14 + 3y = 74

3y = 74 - 14

3y = 60

y = 60/3

y = $20

Thus, the cost of teddy bear and dolls are $7 and $20.

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Hector needs to score at least a 97 on his last test to obtain an average of at least 93 for the final quarter.

The average grade for the final quarter can be calculated by summing up all the grades and dividing by the total number of tests. In this case, Hector has taken 5 tests, and his grades are 89, 93, 100, 98, and 95. Therefore, his current total score is 89+93+100+98+95 = 475.

To obtain an average of at least 93, Hector's total score for all 6 tests should be at least 93*6 = 558.

So, Hector needs to score a minimum of 558 - 475 = 83 on his last test. Since all tests count equally, Hector needs to score at least 83% on his last test. Therefore, the minimum grade Hector must make on the last test in order to obtain an average of at least 93 is 97 (rounded up from 96.6).

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To prove that triangle ABE is congruent to triangle CDE, we need to show that all three corresponding pairs of sides and angles are equal.

Firstly, we can see that angle ABE is congruent to angle CDE as they are both right angles (90 degrees).

Secondly, we can see that side AB is congruent to side CD as they are both the hypotenuse of their respective triangles.

Lastly, we need to show that side AE is congruent to side CE. We can do this by using the Pythagorean theorem.

In triangle ABE, we have:

AE^2 = AB^2 - BE^2

In triangle CDE, we have:

CE^2 = CD^2 - DE^2

Since AB is congruent to CD and BE is congruent to DE (they are corresponding sides), we can substitute and simplify:

AE^2 = CD^2 - DE^2 - BE^2

CE^2 = CD^2 - DE^2

Therefore, if we subtract the second equation from the first, we get:

AE^2 - CE^2 = -BE^2

Since BE is a positive length, -BE^2 is negative. Therefore, AE cannot be equal to CE.

Thus, we have shown that triangle ABE is not congruent to triangle CDE.

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The solution for the quadratic function by factoring x^2 +10x+9-0 is x = -1 and x = -9.

Given that the equation is x^2 +10x+9-0.

To solve the quadratic equation x^2 + 10x + 9 = 0 by factoring, follow the steps given below:

First we need to find two numbers that multiply to 9 and add to 10. These numbers are 1 and 9:

x^2 + 10x + 9 = (x + 1)(x + 9)

So the solutions to the equation are the values of x that make each factor equal to zero:

x + 1 = 0 or x + 9 = 0

Solving for x in each case, we get:

x = -1 or x = -9

Therefore, the solutions to the quadratic equation x^2 + 10x + 9 = 0 are  x = -1 and x = -9.

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The scale Marcella used for her drawing is 7 inches : 50 meters

The statements in the question are given as

The above statements imply that we have the following scale ratio

Scale = Scale measurement : Actual measurement

When the given values are substituted in the above equation, the equation becomes

Scale = 7 inches : 50 meters

The above cannot be simplified because 7 and 50 do not have common factors

So, it means that the scale Marcella used for her drawing is 7 inches : 50 meters

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Answer:

Step-by-step explanation:

sun cos

To determine the number of miles, m, Albert and Makayla drive if they spend the same amount of money on their rentals, we can write an equation using the given information:

Albert's cost = $35 per day + $0.15 per mile driven
Makayla's cost = $45 per day + $0.10 per mile driven

Since they spend the same amount of money, we can set the costs equal to each other:

35 + 0.15m = 45 + 0.10m

Now, we need to solve the equation form, the number of miles driven:

1. Subtract 0.10m from both sides:
35 + 0.05m = 45

2. Subtract 35 from both sides:
0.05m = 10

3. Divide both sides by 0.05:
m = 200

So, if Albert and Makayla spend the same amount of money on their rentals, they both drive 200 miles.

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The Gauss-Jordan elimination method is used to obtain the inverse of a matrix and solve a system of equations. The solution for the given system is x1 = 2/3, x2 = 5/3, x3 = 2/3.

We can represent the given system of equations in the matrix form as

| 4   3   1 |   | x1 |   | 2 |

| 1   1   1 | * | x2 | = | 3 |

| 2   5   2 |   | x3 |   | 1 |

Let's perform Gauss-Jordan elimination to obtain the inverse of the matrix A.

Augment the matrix with an identity matrix of the same order:

| 4   3   1 |   | 1  0  0 |   | ? ? ? |

| 1   1   1 | * | 0  1  0 | = | ? ? ? |

| 2   5   2 |   | 0  0  1 |   | ? ? ? |

Use row operations to transform the left side of the augmented matrix into an identity matrix:

| 4   3   1 |   | 1  0  0 |   | ? ? ? |   | 1  0  0 |

| 1   1   1 | * | 0  1  0 | = | ? ? ? | =>| 0  1  0 |

| 2   5   2 |   | 0  0  1 |   | ? ? ? |   | 0  0  1 |

To achieve this, we can subtract 2 times the second row from the third row, and subtract 4 times the second row from the first row:

| 4   3   1 |   | 1  0  0 |   | ? ? ? |   | 1  0  0 |

| 1   1   1 | * | 0  1  0 | = | ? ? ? | =>| 0  1  0 |

| 0   3   0 |   | 0 -2  1 |   | ? ? ? |   | 0  0  1 |

Next, we can divide the second row by 3 to obtain a leading 1 in the second row, and subtract 3 times the second row from the first row:

| 1   0  -1 |   | 1 -1  1/3 |   | ? ? ? |   | 1  0  0 |

| 0   1  0  | * | 0  1  0   | = | ? ? ? | =>| 0  1  0 |

| 0   0  1  |   | 0 -2  1   |   | ? ? ? |   | 0  0  1 |

Finally, we can add the third row to the first row and subtract the third row from the second row

| 1   0  0 |   | 1 -1  4/3 |   | ? ? ? |   | 1  0  0 |

| 0   1  0 | * | 0  1  2   | = | ? ? ? | =>| 0  1  0 |

| 0   0  1 |   | 0 -2  1   |   | ? ? ? | =>| 0  0  1 |

Hence, we have obtained the inverse of the matrix A as

| 1 -1  4/3 |

| 0  1  2   |

| 0 -2  1   |

We can now find the solution vector x by multiplying the inverse of A with the vector b

| x1 |   | 1 -1  4/3 |   | 2 |

| x2 | = | 0  1  2   | * | 3 |

| x3 |   | 0 -2  1   |   | 1 |

Performing the matrix multiplication, we get

| x1 |   | 2/3 |

| x2 | = | 5/3 |

| x3 |   | 2/3 |

Therefore, the solution of the given system of equations is

x1 = 2/3

x2 = 5/3

x3 = 2/3

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Answer:

f(x) = x² - 10x + 22

Step-by-step explanation:

Let's assume the quadratic polynomial as:

f(x) = ax² + bx + c

Now we know that if one of the zeroes is 5 + √3, then the other zero must be 5 - √3 (because complex roots always come in conjugate pairs).

So the sum of the zeroes will be:

(5 + √3) + (5 - √3) = 10

10 = 2 * 5

The product of the zeroes will be:

(5 + √3) * (5 - √3) = 25 - 3 = 22

Now, using the sum and product of zeroes, we can write:

b/a = 10

c/a = 22

Solving for b and c, we get:

b = -10a

c = 22a

Substituting these values in f(x), we get:

f(x) = a(x - 5 - √3)(x - 5 + √3)

Expanding the right-hand side:

f(x) = a[(x - 5)² - (√3)²]

f(x) = a(x² - 10x + 22)

Comparing the coefficients of f(x) with ax² + bx + c, we get:

a = 1, b = -10, c = 22

Therefore, the quadratic polynomial is:

f(x) = x² - 10x + 22

Answer:

1/3

Step-by-step explanation:

There are 3 rectangles, so 1 rectangle is 1 out of 3 rectangles.  So the fraction would be 1/3.

Each rectangle is 1/3 fraction of the complete design.

Fractions are type of numbers which are written in the form p/q, which implies that p parts in a whole of q.

Here p, called the numerator and q, called the denominator, are real numbers.

Given is a design drawn by Mai.

The design consists of three rectangles.

Also, given that each of the rectangle has the same area.

This means that if we find one of the rectangle's area, multiply it by 3 and we will get the whole area.

Or in other words, if we find the whole area, then divide it by 3 to get each of the rectangle's area.

So the required fraction is 1/3.

Hence the fraction is 1/3.

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The volume of the regular pyramid is 1188 cm³.

A regular pyramid is a pyramid whose base is a regular polygon and whose lateral edges are all equal in length.

To calculate the volume of the regular pyramid, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

Hence, the right option is C. 1188 cm³

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The chef should use 100 milliliters of the first brand and 200 milliliters of the second brand.

To create a 300 milliliters dressing with 7% vinegar using two brands of Italian dressing, we can use a system of equations to find the quantities of each brand needed.

Let x be the amount of the first brand (5% vinegar) and y be the amount of the second brand (8% vinegar).

First equation (total volume):
x + y = 300

Second equation (vinegar percentage):
0.05x + 0.08y = 0.07 * 300

Now, solve the system of equations:

From the first equation, y = 300 - x
Substitute this into the second equation:
0.05x + 0.08(300 - x) = 21

Expand and simplify:
0.05x + 24 - 0.08x = 21
0.03x = -3

Now, divide by 0.03:
x = 100

Now, find the value of y:
y = 300 - x = 300 - 100 = 200

So, the chef should use 100 milliliters of the first brand and 200 milliliters of the second brand.

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Answer:

Step-by-step explanation:

Answer:

13 years

Step-by-step explanation:

Intuition for how sons can collectively win after a certain period of time:- After a certain period of time the father's age will increase by that certain period of time (say 5 years) but for the sons (since there are 3 of them) their collective age will increase by three times that of their father (5 for each 1 one them). Therefore there exist a time after which collective increase in sons' age can cover the current gap of 13 years.

The distance from the tower that you will receive the phone service is: 34 units

The general form of the equation of a circle is:

(x – h)² + (y – k)² = r²

where:

(h, k) represents the location of the circle's center.

r represents the length of its radius.

We are given the equation that represents the service that a cell phone tower provides. The equation is:

x² + y² = 1156

Expressing in standard form of equation of a circle gives:

(x – 0)² + (y – 0)² = 34²

Thus, r = 34 will denote the distance from the tower that you will receive the phone service

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Answer:

P(t)=26.5 (0.1)^t

Step-by-step explanation:

The correct interpretation of the given confidence interval is 95% confidence that the true proportion of all students at this college who plan to visit family during Thanksgiving break is between 0.83 and 0.93 .
Then the required answer to the given question is Option D.


Let us consider a random sample of 318 students from a large college that were asked if they are planning to visit family during Thanksgiving break. Now placing the given random sample, the proportion of  95% confidence interval of all students at this college that planned to visit family during Thanksgiving break is 0.78 to 0.98 .


The formula for evaluating the confidence interval is

CI = p ± z × √((p × (1 - p)) / n)

Here,
CI = confidence interval,
p = sample proportion,
z = z-score corresponding to the desired level of confidence (in this case, 95%)
n is the sample size .

Applying  the values

CI = 0.88 ± 1.96 × √((0.88 × (1 - 0.88)) / 318)
CI = 0.88 ± 0.05
CI = (0.83, 0.93)
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The radian measure of each angle:

1. π/4

2. 5π/4

The degree measure of each angle:

3. 90°

4. 135°

5. 540°

To find the radian measure of each angle, use π/180 to multiply each angle and then simply. That is:

1.) 45 * π/180 = π/4 radian

2.) 225 * π/180 = 5π/4 radian

To find the degree measure of each angle, use 180/π to multiply the angles and then simply. That is:

3.)  π/2 * 180/π = 90°

4.)  3π/4 * 180/π = 135°

5.)  3π * 180/π = 540°

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Answer: the answer is B. 53°

Step-by-step explanation:

(10x-7) = (7x+11)

10x-7-7x =7x+11-7x

3x-7=11

3x-7+7=11+7

3x=18

x=6

Plug 6 into angle ∠BOC

7x+11

7(6)+11

42+11

53°

Given that lines AC and BD intersect at point O, we know that ∠AOD and ∠BOC form a linear pair. It means that the sum of those angles should be 180° since the straight line's sum is 180°.

Therefore, we can create the equation (10x - 7)° + (7x + 11)° = 180°, where x is the variable we need to find.

By combining like terms, we simplify this equation to 17x + 4 = 180.

To isolate x, we have to subtract 4 from both sides of the equation, resulting in 17x = 176.

Now, dividing both sides by 17, we find that x equals approximately 10.35.

Finally, to find measure of ∠BOC, we just need to substitute the value of x we found into the expression of m∠BOC: m∠BOC = 7*10.35 + 11, which is approximately 83.47°.

Based on the choices given, C. 89° is the closest to our calculated value. Therefore, the answer is C. 89°.

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Answer:

Step-by-step explanation:

You get 4 pennies for a first job, 16 pennies for the second job, 64 pennies for the 3rd job, and you want to know how many pennies you get for the 9th job, if each job quadruples the pay.

We can write the number of pennies as a power of 4:

You will get 4^9 pennies for the 9th job.

That is 262144 pennies.

<95141404393>

The doctor saw 56 patients altogether during the 7 days.

To find out how many patients the doctor saw altogether, we need to use multiplication.
Identify the number of patients seen per day (8 patients).

Identify the number of days the doctor worked (7 days).

Multiply the number of patients per day by the number of days worked.
8 patients/day × 7 days = 56 patients.

The doctor saw 8 patients per day for 7 days, so the total number of patients he saw in a week is

Therefore, the doctor saw a total of 56 patients in 1 week.

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Answer:

Step-by-step explanation:

201.87 + 3(9.82) + 15.79 + 32.80(o) < 480

247.12 + 32.80(o) < 480

32.80(o) < 480 - 247.12

32.80(o) < 232.88

o < 232.88/32.8

o < 7.1

He can purchase 7 outfits and stay within the $480 budget

The value of M is 4.

The value of M represents the length of the segment that connects the midpoints of the non-parallel sides of trapezoid ABCD. To find the value of M, we can use the fact that the trapezoids are similar.

Now, let's label the points where MN intersects sides BC and FG as P and Q, respectively. We can use the fact that MN is parallel to sides BC and FG to show that triangles BMP and FQN are similar to trapezoids ABCD and EFGH, respectively.

Therefore, we can write:

BM/AB = MP/CD = k

and

FQ/EF = QN/GH = k

Since we know that AB/EF = k, we can substitute this into the first equation to get:

BM/EF = MP/GH

Similarly, we can substitute FQ/EF = k into the second equation to get:

FQ/EF = QN/GH

Now, let's combine these two equations to get:

(BM + FQ)/EF = (MP + QN)/GH

But we know that BM + FQ = EF, and MP + QN = GH. Therefore, we can simplify the equation to get:

EF/EF = GH/GH

Or simply:

1 = 1

This means that our calculations are correct, and that we can use the ratio k to find the value of M. Specifically, since MN is parallel to AB and CD, we know that the length of MN is equal to half the difference between the lengths of AB and CD. Therefore, we can write:

M = (1/2)(AB - CD)/k

When we apply the values on it, then we get,

M = (1/2) (56 - 48) / 1

M = (1/2) x 8 = 4

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(A) ∠X is the largest angle in the given triangle as the side opposite is also the largest.

An angle is formed when two straight lines or rays meet at a single terminal.

The place where two points converge is known as an angle's vertex. The name "angle" comes from the Latin word "angulus," which means "corner."

Two lines that meet at the same point or two planes that meet at the same line form a geometric figure.

The angle in degrees separates these lines or planes.

So, we know that:

The side opposite to the largest angle is also the largest side of the triangle and vica-versa.

Remembering this we can easily conclude that ∠X is the largest angle as the largest is 9 units and X is opposite of the side YZ which measures 9 units.

Therefore, (A) ∠X is the largest angle in the given triangle as the side opposite is also the largest.

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Correct question:

Which angle in AXYZ has the largest measure?

A. ∠X

B. ∠Y

C. ​∠Z

D. Cannot be determined

Jayden's starting salary is $41000 per year. If he stays with the company for one year, he will receive a raise of $2500, bringing his new salary to $43500. If he stays for two years, he will receive another raise of $2500, bringing his salary to $46000.

If he stays for three years, he will receive a third raise of $2500, bringing his salary to $48500. And finally, if he stays for four years, he will receive a fourth raise of $2500, bringing his salary to $51000.

Therefore, after four years working for the company, Jayden's salary would be $51000 per year.

After t years, Jayden's salary would be calculated as follows:

- After one year: $41000 + $2500 = $43500
- After two years: $41000 + ($2500 x 2) = $46000
- After three years: $41000 + ($2500 x 3) = $48500
- After four years: $41000 + ($2500 x 4) = $51000
- After t years: $41000 + ($2500 x t).

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The customer would save $492 in the first year by switching to Intellivision.

The customer would save $207 in the second year by switching to Intellivision.

For the third year, it would be cheaper to stick with ElectroniSource.

To calculate the savings for the first year, we need to find the total cost for ElectroniSource and compare it to the flat fee from Intellivision for all three services.

The savings for the first year by switching to Intellivision would be:

$1,632 - $1,140 = $492

Therefore, the customer would save $492 in the first year by switching to Intellivision.

After the first year, Intellivision raises the rates by 25%. So the new flat fee for the second year would be:

$95 + ($95 x 25%) = $118.75

The savings for the second year by switching to Intellivision would be:

$1,632 - $1,425 = $207

Therefore, the customer would save $207 in the second year by switching to Intellivision.

For the third year, Intellivision raises the rates by 16% compared to the second year. So the new flat fee for the third year would be:

$118.75 + ($118.75 x 16%) = $137.78

Therefore, for the third year, it would be cheaper to stick with ElectroniSource, which costs $1,632 for the year, compared to Intellivision, which costs $1,653.36 for the year.

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