The upper-left coordinates on a rectangle are ( − 1 , 7 ) (−1,7)left parenthesis, minus, 1, comma, 7, right parenthesis, and the upper-right coordinates are ( 4 , 7 ) (4,7)left parenthesis, 4, comma, 7, right parenthesis. The rectangle has an area of 20 2020 square units.

The dairy farmer should use 400 pounds of the 75% protein supplement and 800 pounds of the 30% protein ration to make 1200 pounds of a high-grade 45% protein ration.

To determine how many pounds of each protein supplement the dairy farmer should use, we can set up a system of equations based on the given information.

Let's denote the following:

x = pounds of the 75% protein supplement

y = pounds of the 30% protein ration

The total weight of the mixture is 1200 pounds, so we have the equation:

x + y = 1200

The protein content in the mixture should be 45% of the total weight, so we have the equation:

0.75x + 0.30y = 0.45 * 1200

Now we can solve this system of equations.

First, let's rewrite the second equation in decimal form:

0.75x + 0.30y = 540

To make it easier to work with, we can eliminate the decimals by multiplying both sides of the equation by 100:

75x + 30y = 54000

Now we have a system of equations:

x + y = 1200

75x + 30y = 54000

We can solve this system using any appropriate method, such as substitution or elimination. In this case, let's use the substitution method:

Solve the first equation for x:

x = 1200 - y

Substitute the value of x into the second equation:

75(1200 - y) + 30y = 54000

Simplify and solve for y:

90000 - 75y + 30y = 54000

-45y = -36000

y = 800

Now we can substitute the value of y back into the first equation to find x:

x + 800 = 1200

x = 1200 - 800

x = 400

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The expression X12 + y11 x W3 represents the sum of the variable X12 and the product of y11 and W3.

To evaluate the expression X12 + y11 x W3,

1. Determine the value of X12: If X12 is a known constant or variable, substitute its value into the expression. If it is unknown, the expression remains as X12.

2. Determine the value of y11: If y11 is a known constant or variable, substitute its value into the expression. If it is unknown, the expression remains as y11.

3. Determine the value of W3: If W3 is a known constant or variable, substitute its value into the expression. If it is unknown, the expression remains as W3.

4. Multiply y11 by W3: Multiply the values of y11 and W3 together to obtain the product.

5. Add X12 to the product: Add the value of X12 (or the expression itself if it is unknown) to the product obtained in the previous step.

6. Simplify the expression: Combine like terms, if applicable, to simplify the expression further.

By following these steps, you can evaluate the given expression X12 + y11 x W3 and obtain the final answer. Remember to substitute known values and simplify the expression whenever possible.

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According to the information, the number of the image would be 17/100 or 0.17

To express this number in fraction and decimal we must take into account different aspects. First of all we must consider the reference numbers that are on the number line.

On the other hand, we must count how many segments each unit of the line is divided into. In this case it is in 10. So to identify the number we can add a 0 to the denominator and take the numerator as an integer that would be 17. Then the fraction would remain:

In the case of the decimal we only have to carry out the division and we will obtain the decimal.

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The angles of the cyclic quadrilateral PQRS and the angles of the triangle XSR indicates that the measure of the external angle to the triangle XSR, P[tex]\widehat{X}[/tex]R is 120°

A cyclic quadrilateral is a quadrilateral inscribed in a circle.

Whereby the required value is the measure of the angle P[tex]\widehat{X}[/tex]R, the quadrilateral QPRS indicates that we get;

The properties of cyclic quadrilaterals indicates; angles ∠PQR and ∠PSR are supplementary angles, therefore;

m∠PQR + m∠PSR = 180°

Therefore; 80° + m∠PSR = 180° (Substitution property)

m∠PSR = 180° - 80° = 100°

m∠PSR = 100°

The external angle of a triangle ΔXSR indicates;

P[tex]\widehat{X}[/tex]R = m∠PSR + m∠SRT

Therefore; P[tex]\widehat{X}[/tex]R = 100° + 20° = 120°

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To calculate the circumference of a circle, you can use the formula:

[tex]\displaystyle C=2\pi r[/tex]

Where [tex]\displaystyle C[/tex] represents the circumference and [tex]\displaystyle r[/tex] represents the radius of the circle.

Given that the radius [tex]\displaystyle r[/tex] is 8 inches, we can substitute this value into the formula:

[tex]\displaystyle C=2\pi (8)[/tex]

Simplifying the expression:

[tex]\displaystyle C=16\pi [/tex]

Thus, the circumference of a circle with a radius of 8 inches is [tex]\displaystyle 16\pi [/tex] inches.

Note: [tex]\displaystyle \pi [/tex] represents the mathematical constant pi, which is approximately equal to 3.14159.

[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]

♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

Answer:

15

Step-by-step explanation:

180-15

Hello!

the sum of the angles in the triangle = 180°

so the 3rd angle = 180° - 165° = 15°

Answer:

The formula for finding the area of a circle using its radius is:

The formula for finding the area of a circle using its radius is:A = πr²

The formula for finding the area of a circle using its radius is:A = πr²where A is the area of the circle and r is its radius.

The formula for finding the area of a circle using its radius is:A = πr²where A is the area of the circle and r is its radius.The formula for finding the area of a circle using its diameter is:

The formula for finding the area of a circle using its radius is:A = πr²where A is the area of the circle and r is its radius.The formula for finding the area of a circle using its diameter is:A = (π/4)d²

The formula for finding the area of a circle using its radius is:A = πr²where A is the area of the circle and r is its radius.The formula for finding the area of a circle using its diameter is:A = (π/4)d²where A is the area of the circle and d is its diameter.

The formula for finding the area of a circle using its radius is:A = πr²where A is the area of the circle and r is its radius.The formula for finding the area of a circle using its diameter is:A = (π/4)d²where A is the area of the circle and d is its diameter.Note that π (pi) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter and is approximately equal to 3.14

Step-by-step explanation:

Hope it Helps

The formula for finding the area of a circle using its radius is π * r².

The formula for finding the area of a circle using its diameter is (π/4) * d².

The formula for finding the area of a circle using its radius is:

A = π * r²

where:

A represents the area of the circle,

π (pi) is a mathematical constant approximately equal to 3.14159, and

r represents the radius of the circle.

If you have the diameter of the circle instead of the radius, you can use the following formula:

A = (π/4) * d²

where:

A represents the area of the circle,

π (pi) is a mathematical constant approximately equal to 3.14159, and

d is the diameter of the circle.

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The three-digit number with the given conditions is 309.

Let's assume the hundreds digit of the three-digit number is "x". Since the tens digit is three less than the hundreds digit, the tens digit can be represented as "x - 3". Similarly, the unit digit is three times the hundreds digit, so it can be represented as "3x".

According to the given information, the sum of the digits of the three-digit number is 12. Therefore, we can write the equation:

x + (x - 3) + 3x = 12

Combining like terms:

5x - 3 = 12

Adding 3 to both sides:

5x = 15

Dividing both sides by 5:

x = 3

So, the hundreds digit of the number is 3. Substituting this value back into the expressions for the tens and unit digits:

Tens digit = x - 3 = 3 - 3 = 0

Unit digit = 3x = 3 * 3 = 9

Therefore, the three-digit number with the given conditions is 309.

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The area of the composite figure is 800m²

The area of a figure is the number of unit squares that cover the surface of a closed figure.

Composite geometric figures are made from two or more geometric figures.

The figure consist of a rectangle , a semi circle and a triangle.

Area of the semicircle = 1/2 πr²

= 1/2 × 3.14 × 10²

= 314/2 = 157 m²

Area of the rectangle = l × w

= 25 × 20

= 500m²

area of the triangle = 1/2bh

= 1/2 × 10 × 25

= 25 × 5

= 125 m²

Therefore the area of the composite figure

= 125 + 500 + 175

= 800m²

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The value of cos(45°) is √2/2. The correct answer choice is D. √2.

In the given diagram, the angle labeled as 45° is part of a right triangle. To find the value of cos(45°), we need to determine the ratio of the adjacent side to the hypotenuse.

Since the angle is 45°, we can assume that the triangle is an isosceles right triangle, meaning the two legs are congruent. Let's assume the length of one leg is x. Then, by the Pythagorean theorem, the length of the hypotenuse would be x√2.

Now, using the definition of cosine, which is adjacent/hypotenuse, we can substitute the values:

cos(45°) = x/(x√2) = 1/√2

Simplifying further, we rationalize the denominator:

cos(45°) = 1/√2 * √2/√2 = √2/2

Therefore, the value of cos(45°) is √2/2.

The correct answer choice is D. √2.

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Considering the definition of probability, the probability that the gumball machine dispenses 2 red gumballs in a row is 7/59.

Probability establishes a relationship between the number of favorable events and the total number of possible events.

The probability of any event A is defined as the ratio between the number of favorable cases (number of cases in which event A may or may not occur) and the total number of possible cases. This is called Laplace's Law.

P(A)= number of favorable cases÷ number of posible cases

The conditional probability P(A|B) is the probability that event A occurs, given that another event B also occurs. That is, it is the probability that event A occurs if event B has occurred. It is defined as:

P(A|B) = P(A∩B)÷ P(B)

Expressed another way:

P(A∩B)= P(B)× P(A|B)

In this case, you know a gumball machine has:

Considering the definition of probability, since there are 21 red gumballs in a total of 60 gumballs, the probability of getting a red gumball on the first try is 21/60.

Now, the probability of getting a red gumball on the second try, given that you got a red gumball on the first try and now you have a 59 gumballs an 20 red gumballs on the machine, is 20/59.

So, the probability of getting 2 red gumballs in a row is calculated as:

P(A∩B)= P(B)× P(A|B)

P(A∩B)= 21/60× 20/59

P(A∩B)= 7/59

Finally, the probability that the gumball machine dispenses 2 red gumballs in a row is 7/59.

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Answer:

Step-by-step explanation:

y

=

1

3

x

−

3

Use the slope-intercept form to find the slope and y-intercept

Slope:

1

3

y-intercept:

(

0

,

−

3

)

Any line can be graphed using two points. Select two

x

values, and plug them into the equation to find the corresponding

y

values.

x

y

0

−

3

3

−

2

Graph the line using the slope and the y-intercept, or the points.

Slope:

1

3

y-intercept:

(

0

,

−

3

)

x

y

0

−

3

3

−

2

image of graph

Answer:

a)The upper quartile of the discounts is £40

The lower quartile of the discounts is £25

b)The interquartile range is £15

Step-by-step explanation:

a) Upper and lower quartiles,

The upper quartile contains 75% or 3/4 of the total number of values,

In our case, since there are 80 items ,so 3/4 of 80 is, (3/4)(80) = 60

Hence for the upper quartile, the cumulative frequency should be 60, which gives a discount value of £40. (60 items have a discount less than or equal to £40)

So, The upper quartile of the discounts is £40

The lower quartile contains 25% or 1/4 of the values,

So, In our case, 1/4 of 80 is, (1/4)(80) = 20

Hence for the lower quartile, the cumulative frequency should be 20,

which gives a discount value of £25

So, the lower quartile of the discounts is £25

b) The interquartile range,

The interquartile range is the difference between the upper and lower quartiles,

so,

interquartile range = upper quartile - lower quartile

interquartile range = 40 - 25

interquartile range = £15

Hence, the interquartile range is £15

a) The price elasticity of demand is -2.

b) It  means that a decrease in price from 400 birr to 200 birr results in a relatively larger increase in the quantity demanded from 2000 to 4000.

To find the price elasticity of demand, we need to determine the percentage change in quantity demanded divided by the percentage change in price.

(a) Price elasticity of demand (ε) can be calculated using the following formula:

ε = (ΔQ/Q) / (ΔP/P)

Where:

ΔQ = Change in quantity demanded

Q = Initial quantity demanded

ΔP = Change in price

P = Initial price

In this case, when the price changes from 400 birr to 200 birr, the quantity demanded changes from 2000 to 4000. Let's calculate the percentage changes:

ΔQ/Q = (4000 - 2000)/2000 = 2000/2000 = 1

ΔP/P = (200 - 400)/400 = -200/400 = -0.5

Now we can substitute these values into the formula:

ε = (1)/(-0.5) = -2

The price elasticity of demand is -2.

(b) Based on the value of the price elasticity of demand (-2), we can determine the type of elasticity:

If the price elasticity of demand is greater than 1, it is considered elastic demand. This means that a small change in price will result in a relatively large change in quantity demanded. In this case, the demand is responsive to price changes.

Since the price elasticity of demand is -2, which is greater than 1, we can conclude that the demand for the concert tickets is elastic. This means that a decrease in price from 400 birr to 200 birr results in a relatively larger increase in the quantity demanded from 2000 to 4000.

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Answer: So, the sales tax on the item is $49.

Step-by-step explanation:

The sales tax rate is already given as 14%. It is stated that the item costs $350 before tax, and the sales tax rate is 14%. Therefore, the sales tax amount can be calculated by multiplying the cost of the item by the tax rate:

Sales tax amount = $350 * 14% = $350 * 0.14 = $49

So, the sales tax on the item is $49.

Answer:

FH = 35.64

Step-by-step explanation:

(∠A = 360 so the other angle is 40)

By law of cosines,

FH² = AH² + FA² - 2(AH)(FA) * cos(A)

= 30² + 25² - 2(30)(25) * cos(80)

= 900 + 625 - 1500 * 0.17

= 1525 - 255

FH²  = 1270

FH = √1270

FH = 35.64

The height of the hill is approximately 25.73 ft. Where v0 is the initial velocity (108 ft/s), g is the acceleration due to gravity [tex](-32.2 ft/s^2)[/tex],

To find the height of the hill, we can use the formula for the vertical position of an object under constant acceleration:

h = h0 + v0t + 1/2at^2

where h is the final height, h0 is the initial height, v0 is the initial velocity, t is the time, and a is the acceleration due to gravity (-32.2 ft/s^2).

In this case, we are given that the initial height h0 is 3 ft, the initial velocity v0 is 108 ft/s, and the time t is 9.5 s. We want to find the height of the hill, which we can denote as h_hill. The final height is the height of the plain, which we can denote as h_plain and assume is zero.

At the highest point of its trajectory, the arrow will have zero vertical velocity, since it will have stopped rising and just started to fall. So we can set the velocity to zero and solve for the time it takes for that to occur. Using the formula for velocity under constant acceleration:

v = v0 + at

we can solve for t when v = 0, h0 = 3 ft, v0 = 108 ft/s, and a = -32.2 ft/s^2:

0 = 108 - 32.2t

t = 108/32.2 ≈ 3.35 s

Thus, it takes the arrow approximately 3.35 s to reach the top of its trajectory.

Using the formula for the height of an object at a given time, we can find the height of the hill by subtracting the height of the arrow at the top of its trajectory from the initial height:

h_hill = h0 + v0t + 1/2at^2 - h_top

where h_top is the height of the arrow at the top of its trajectory. We can find h_top using the formula for the height of an object at the maximum height of its trajectory:

h_top = h0 + v0^2/2a

Plugging in the given values, we get:

h_top = 3 + (108^2)/(2*(-32.2)) ≈ 196.78 ft

Plugging this into the first equation, we get:

h_hill = 3 + 108(3.35) + 1/2(-32.2)(3.35)^2 - 196.78

h_hill ≈ 25.73 ft

If the arrow is launched at t0, the equation describing velocity as a function of time would be:

v(t) = v0 - gt

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To find the measures of angles ABD and DBC in the right triangle ABC, we considered two cases: when point D lies on segment AB and when point D lies on segment BC. In both cases, the measures of the angles were determined based on the given ratio of 3:2.

In the given problem, we have a right triangle ABC, and we need to find the measures of angle ABD (mABD) and angle DBC (mDBC) based on the ratio of the measures of angle A and angle B.

Let's consider the different cases for the location of point D:

Case 1: Point D lies on segment AB.

In this case, angle ABD and angle DBC will be acute angles. Let's assume that angle ABD has a measure of 3x and angle DBC has a measure of 2x. Since angle ABD and angle DBC are acute angles, the sum of their measures should be less than 90 degrees.

Therefore, we have the inequality: 3x + 2x < 90. Solving this inequality, we get 5x < 90, which gives x < 18. So, the measure of angle ABD (mABD) will be 3x, and the measure of angle DBC (mDBC) will be 2x.

Case 2: Point D lies on segment BC.

In this case, angle ABD and angle DBC will be obtuse angles. Let's assume that angle ABD has a measure of 3x and angle DBC has a measure of 2x. Since angle ABD and angle DBC are obtuse angles, the sum of their measures should be greater than 90 degrees.

Therefore, we have the inequality: 3x + 2x > 90. Solving this inequality, we get 5x > 90, which gives x > 18. So, the measure of angle ABD (mABD) will be 3x, and the measure of angle DBC (mDBC) will be 2x.

By investigating these two cases, we can find the measures of angle ABD (mABD) and angle DBC (mDBC) based on the given ratio of 3:2. The specific values of mABD and mDBC will depend on the exact location of point D within the triangle ABC.

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1. The fundamental undefined concepts in Geometry are point, line, and plane.

2. A line is formed by connecting at least two points.

3. Collinear points are three or more points that lie on the same line.

4. Concurrent lines are three or more lines that intersect at a common point.

5. A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side.

6. An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side or its extension.

7. The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect.

8. The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect.

9.The incenter is the point where the angle bisectors of a triangle intersect.

10. The orthocenter is the point where the altitudes of a triangle intersect.

1. The three fundamental Undefined Concepts in Geometry are:

Point: A location in space that has no size or dimension.

Line: A straight path that extends infinitely in both directions.

Plane: A flat, two-dimensional surface that extends infinitely in all directions.

2. A Line consists of at least two points: A line is determined by two distinct points and extends infinitely in both directions.

3. A Plane consists of at least three non-collinear points: A plane is determined by three non-collinear points and extends indefinitely in all directions.

4. . Three or more points that lie on one line are called collinear: Collinear points are points that lie on the same straight line.

5. . Three or more lines that pass through one point are called concurrent: Concurrent lines are lines that intersect at a common point.

6.A line segment joining a vertex of a triangle to the mid-point of the opposite side is called a median: The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side.

7. A line segment drawn from a vertex of a triangle perpendicular to the opposite side (or to the opposite side produced) is called an altitude: The altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side or its extension.

8. The perpendicular bisectors of the sides of any triangle are concurrent at a point which is equidistant from the vertices of the triangle: The point of concurrency of the perpendicular bisectors of the sides of a triangle is equidistant from the triangle's vertices.

9. The angle bisectors of any triangle are concurrent at a point which is equidistant from the sides of the triangle: The point of concurrency of the angle bisectors of a triangle is equidistant from the triangle's sides.

10. The point of intersection of the altitudes of a triangle is called the orthocenter: The orthocenter is the point of intersection of the altitudes of a triangle.

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Answer:

5.44 meters

Step-by-step explanation:

We Know

0.3048 meter = 1 ft

How many ft makes a height of 1.66m?

We Take

1.66 ÷ 0.3048 ≈ 5.44 meters

So, the answer is 5.44 meters.

Answer:

5.44 meters

Step-by-step explanation:

We Know

0.3048 meter = 1 ft

How many ft makes a height of 1.66m?

We Take

1.66 ÷ 0.3048 ≈ 5.44 meters

So, the answer is 5.44 meters

If he uses 400 ml of milk, the grams of raisins Stan would need is 8 g of raisins.

In Mathematics and Geometry, a direct proportion simply refers to a mathematical equation which is typically used to represent the equality of two (2) ratios.

By applying direct proportion to the given information, we have the following mathematical equation:

200 ml of milk = 40 g of raisins

400 ml of milk = x g of raisins

By cross-multiplying, we have the following:

200x = 40 × 400

200x = 1600

x = 1600/200

x = 8 g of raisins.

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Answer:

The lower quartile of the fish lengths is 15 inches

The upper quartile of the fish lengths is 30 inches

Step-by-step explanation:

(the cumulative frequency gives how much of the observations/measurements are under a certain value e.g, in our case, for lengths of 40 inches, all 16 of the fish have length in that range (less than or equal to 40) so, the cumulative frequency is 16 and so on)

We need to find the lower and upper quartiles of the fish lengths,

Now, the lower quartile contains 25% (or 1/4) of the values

in our case, since there are 16 fish,

25% of 16 is 4

So, for the lower quartile, the cumulative frequency should be 4.

For which the length is 15 inches,

So, the lower quartile of the fish lengths is 15 inches

The upper quartile contains 75% (or 3/4) of the values,

Now, 3/4 of 16 is 12.

So, for the upper quartile, the cumulative frequency should be 12.

For which the length is 30 inches

So, the upper quartile of the fish lengths is 30 inches

Answer:

Step-by-step explanation:

[tex]\frac{d}{dx} [cos(3x^2) \sqrt[3]{5x^3 -1} +sin(4x^3)]^4\\\\=4[cos(3x^2) \sqrt[3]{5x^3 -1} +sin(4x^3)]^3\; \frac{d}{dx} [cos(3x^2) \sqrt[3]{5x^3 -1} +sin(4x^3)] --- eq(1)[/tex]

Lets look at the derivative part:

[tex]\frac{d}{dx} [cos(3x^2) \sqrt[3]{5x^3 -1} +sin(4x^3)] \\\\= \frac{d}{dx}[cos(3x^2) \sqrt[3]{5x^3 -1} ] + \frac{d}{dx}[sin(4x^3)]\\\\=cos(3x^2) \frac{d}{dx}[ \sqrt[3]{5x^3 -1} ] + \sqrt[3]{5x^3 -1}\frac{d}{dx}[ cos(3x^2) ] + cos(4x^3) \frac{d}{dx}[4x^3]\\\\=cos(3x^2) \frac{1}{3} (5x^3 -1)^{\frac{1}{3} -1} \frac{d}{dx}[5x^3 -1] + \sqrt[3]{5x^3 -1} (-sin(3x^2))\frac{d}{dx}[ 3x^2] + cos(4x^3)[(4)(3)x^2][/tex]

[tex]=\frac{cos(3x^2) 5(3)x^2}{3(5x^3 - 1)^{\frac{2}{3} }} -\sqrt[3]{5x^3 -1}\; sin(3x^2) (3)(2)x + 12x^2 cos(4x^3)\\\\=\frac{5x^2cos(3x^2) }{(5x^3 - 1)^{\frac{2}{3} }} -6x\sqrt[3]{5x^3 -1}\; sin(3x^2) + 12x^2 cos(4x^3)[/tex]

Substituting in eq(1), we have:

[tex]\frac{d}{dx} [cos(3x^2) \sqrt[3]{5x^3 -1} +sin(4x^3)]^4\\\\=4[cos(3x^2) \sqrt[3]{5x^3 -1} +sin(4x^3)]^3\; [\frac{5x^2cos(3x^2) }{(5x^3 - 1)^{\frac{2}{3} }} -6x\sqrt[3]{5x^3 -1}\; sin(3x^2) + 12x^2 cos(4x^3)][/tex]

The length of the rectangle is twice its width. If its perimeter is 7 1/3 cm, its length will be 22/9 cm, and the width is 11/9 cm.

Let's assume the width of the rectangle is "b" cm.

According to the given information, the length of the rectangle is twice its width, so the length would be "2b" cm.

The formula for the perimeter of a rectangle is given by:

Perimeter = 2 * (length + width)

Substituting the given perimeter value, we have:

7 1/3 cm = 2 * (2b + b)

To simplify the calculation, let's convert 7 1/3 to an improper fraction:

7 1/3 = (3*7 + 1)/3 = 22/3

Rewriting the equation:

22/3 = 2 * (3b)

Simplifying further:

22/3 = 6b

To solve for "b," we can divide both sides by 6:

b = (22/3) / 6 = 22/18 = 11/9 cm

Therefore, the width of the rectangle is 11/9 cm.

To find the length, we can substitute the width back into the equation:

Length = 2b = 2 * (11/9) = 22/9 cm

So, the length of the rectangle is 22/9 cm, and the width is 11/9 cm.

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To show that the point (å, ä) is on the perpendicular bisector of the line segment with endpoints (Ů, ü) and (ĝ, ġ), we need to demonstrate two things: that the point lies on the line segment, and that it is equidistant from the endpoints.

1. Determine the midpoint of the line segment:

  - The midpoint coordinates ([tex]x_{mid, y_{mid[/tex]) can be found using the midpoint formula:

    [tex]x_{mid[/tex] = (x1 + x2) / 2 and [tex]y_mid[/tex] = (y1 + y2) / 2, where (x1, y1) and (x2, y2) are the coordinates of the endpoints.

    In this case, we have (x1, y1) = (Ů, ü) and (x2, y2) = (ĝ, ġ).

2. Calculate the midpoint coordinates:

  - Substitute the values into the midpoint formula to find (x_mid, y_mid).

3. Find the slope of the line segment:

  - Use the slope formula: slope = (y2 - y1) / (x2 - x1).

    Apply the formula to the endpoints (Ů, ü) and (ĝ, ġ) to determine the slope of the line segment.

4. Determine the negative reciprocal of the line segment's slope:

  - Take the negative reciprocal of the slope calculated in the previous step. The negative reciprocal of a slope m is -1/m.

5. Write the equation of the perpendicular bisector:

  - Using the negative reciprocal slope and the midpoint coordinates ([tex]x_{mid[/tex], [tex]y_{mid[/tex]), write the equation of the perpendicular bisector in point-slope form: y - [tex]y_{mid[/tex] = [tex]m_{perp[/tex] * (x - [tex]x_{mid[/tex]), where [tex]m_{perp[/tex] is the negative reciprocal slope.

6. Substitute the point (å, ä) into the equation:

  - Replace x and y in the equation of the perpendicular bisector with the coordinates of the point (å, ä). Simplify the equation.

7. Verify that the equation holds true:

  - If the equation is satisfied when substituting (å, ä), then the point lies on the perpendicular bisector.

By following these steps, you can demonstrate that the point (å, ä) lies on the perpendicular bisector of the line segment with endpoints (Ů, ü) and (ĝ, ġ).

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The distance between point P and line L is 16/9√(13).

To find the distance between point P and line L, we can use the formula for the distance between a point and a line in two-dimensional space. The formula is as follows:

Let P = (x1, y1) be the point and L be the line ax + by + c = 0. Then the distance between P and L is:

|ax1 + by1 + c|/√(a² + b²)

To find a, b, and c for the given line, we need to put it in slope-intercept form y = mx + b by solving for y.

2x - 3y = 12=> 2x - 12 = 3y=> (2/3)x - 4 = y

The slope of the line, m, is the coefficient of x, which is 2/3. Therefore, the line is:

y = (2/3)x - 4The values of a, b, and c are: a = 2/3b = -1c = -4

Now we can substitute the coordinates of P and the values of a, b, and c into the formula for the distance between a point and a line.

Let P = (3, 5).|a(3) + b(5) + c|/√(a² + b²)= |(2/3)(3) - 1(5) - 4|/√[(2/3)² + (-1)²]= |-4/3 - 4|/√(4/9 + 1)= 16/9√(13).

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Therefore, the longest length of pencil that John can hide in the pencil holder is approximately 127.32 cm (rounded to two decimal places).

Given that John has a 100 cm³ pencil holder, the longest length of pencil he can hide in the pencil holder can be found by using the formula for the volume of a cylinder which is given by: V = πr²h where r is the radius of the cylinder and h is the height of the cylinder.The volume of a cylinder can also be expressed as V = lwh where l is the length of the cylinder, w is the width of the cylinder and h is the height of the cylinder.

If we assume that the pencil is cylindrical in shape, then its volume can be given by V = πr²l where r is the radius of the pencil and l is the length of the pencil.Since the volume of the pencil holder is given to be 100 cm³, then we have:100 = πr²h (Equation 1)

Now, we need to find the value of l that can fit inside the pencil holder. We can use the formula for the volume of the pencil to do this as follows:

V = πr²lLet V = 100 cm³ and

r = 0.5 cm

(since the pencil holder is cylindrical in shape, we can assume that it has a radius of 0.5 cm)Then, we have:100 = π(0.5)²lSolving for l, we get:

l = 100 / (π(0.5)²)

l = 100 / (0.7854)l ≈ 127.32 cm

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The height of the canvas is approximately 11.5 inches.

To find the height of the canvas, we can use trigonometry and the given angle. Let's denote the height as 'h' inches.

We know that the width of the canvas is 20 inches, and it forms a 35° angle with the diagonal. Since a rectangle has 90° angles, the diagonal forms a right triangle with the width and height of the canvas.

The diagonal can be found using the Pythagorean theorem: diagonal^2 = width^2 + height^2.

In this case, diagonal^2 = 20^2 + h^2.

Since the angle between the width and the diagonal is given as 35°, we can use the sine function: sin(35°) = opposite/hypotenuse = h/diagonal.

Rearranging the equation, we have h = diagonal * sin(35°).

Substituting the value of diagonal^2 from the Pythagorean theorem, we get h = sqrt(20^2 + h^2) * sin(35°).

Solving this equation, we find that h ≈ 11.5 inches.

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