Why is the answer A?

11°

The solution is in the attachment

The angle of subtends the hill will make with horizontal is 17.04 degrees.

The trigonometric functions found in the four quadrants, as well as their graphs, domains, and differentiation and integration, will all be understood.

The trigonometric function is very good and useful in real-life problems.

Given below is the image of the situation,

In triangle ABC →

Tan25° = (125+240)/x

x = 782.745 feet

Now in triangle ADC →

Tan(y) = 240/x

Tan y = 240/782.745

y = 17.04°

Hence "The angle of subtends the hill will make with horizontal is 17.04 degrees".

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Answer:

10.2 feet.

Step-by-step explanation:

We have been given that Derek places  a standard 12-inch ruler next to the goal post and measures the shadow of the ruler and the backboard. If the ruler has a shadow of 10 inches. We are asked t find the height of the backboard, if the backboard has a shadow of 8.5 feet.

We will use proportions to solve our given problem as ratio between sides ruler will be equal to ratio of sides of background.

[tex]\frac{\text{Actual height of ruler}}{\text{Shadow of ruler}}=\frac{\text{Actual height of backboard}}{\text{Shadow of backboard}}[/tex]

[tex]\frac{12}{10}=\frac{\text{Actual height of backboard}}{8.5}[/tex]

[tex]\frac{12}{10}*8.5=\frac{\text{Actual height of backboard}}{8.5}*8.5[/tex]

[tex]1.2*8.5=\text{Actual height of backboard}[/tex]

[tex]10.2=\text{Actual height of backboard}[/tex]

Therefore, the actual height of the back-board is 10.2 feet.

Answer:

Step-by-step explanation:

Since the backboard and the ruler are both vertical and the sun is at the same position in the sky, the triangle made by the backboard and its shadow is similar to the triangle made by the ruler and its shadow.

The ratio of the corresponding sides of the triangles are equal and the height of the backboard can be determined by solving the following proportion.

Therefore, the top of the backboard is 7.64 feet high.

Answer:

         [tex]\large\boxed{\large\boxed{3\times 10^8meters}}[/tex]

Explanation:

One of the applications of scientific notation is to make estimations rounding the whole part of the numbers, i.e. the digits before the decimal point, to the nearest integer, and adding the exponents of the powers of base 10.

Here, you must estimate this product of two numbers written is scientific notation:

          [tex](3.21\times 10^3)(1.13\times 10^5)[/tex]

Then, for an estimation you round 3.21 to 3 and 1.13 to 1, then multiply 3 × 1 = 3. That will be the coefficient of your new power of 10.

The power or exponent will be the sum of the powers of the numbers that are being multiplied, i.e. 3 + 5 = 8.

And the result is [tex]3\times 10^8[/tex]

The unit is meters, so you write your answer as: [tex]3\times 10^8meters[/tex]

Answer:

b. 3*10^8

Step-by-step explanation:

The recommended number of classes is 5, and the class interval is 10. The first class's lower limit would be 50. To create a relative frequency distribution, divide each class frequency by the total

First, we have to calculate the range of the data by subtracting the smallest number from the highest number (98-51 = 47). A typical choice for the number of classes might be between five and 20. The

square root rule

suggests a round number near the square root of the number of observations, which is approx 4.47 in this case; you can round to 5. Now we divide the range by the number of classes (47/5 approximately 9). So, the class interval would be 10 (we round up as it does not make sense to have a fraction of an oil change). If the smallest number is 51, the lower limit for the first class would be 50. Hence, the

frequency distribution

would break down like this: 50-59, 60-69, 70-79, 80-89, 90-99. To create a relative frequency distribution, divide each class frequency by the total number of observations and multiply by 100 to convert to percentages. As for the shape, you would need to plot the data to see, but it may have a normal distribution as service levels often do.

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Answer:

m<7=65

m<4 = 115

m<6 = 115

m<1=65

m<16 = 60

m<18 = 60

m<21 = 120

m<10 = 55

m<11 = 125

m<12 = 55

Step-by-step explanation:

(1) m<7 = 65° (corresponding angles are equal)

(2) m<4 = 180 - 65 = 115 (angles on a straight line)

(3) m<6 = 115 (angles on a straight line)

(4) m<1 = 180-115 = 65

(5) m<16 = 180 - 120 = 60°

(6) m<18 = m<14 = 60°(corresponding angles are equal)

(7) m<21 = 180 - 60 = 120

(8) m<10 = m<12 = 180-(65+60)=55(sum of angles in a triangle)

m<10 = 55°

(9) m<11 = 180 - 55 = 125°

(10)m<12 = 55°(sum of angles in a triangle)

Answer:

Step-by-step explanation:

Answer: In total Katy earned $190

Katy earn  $190 in all on Friday, Saturday, and Sunday.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS is; Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

Given that Katy earns $10 per hour. She worked 4 hours on Friday, 9 hours on Saturday, and 6 hours on Sunday. ​

Therefore, total time = 4+9+6=19

Then 19 x 10 = 190

so, she made $190

Hence, Katy earn  $190 in all on Friday, Saturday, and Sunday.

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Answer: option C is the correct answer

Step-by-step explanation:

Let x represent the cost of one night at the hotel.

Let y represent the cost of renting the car for one day.

Connor went to New York for vacation. He spent 3 nights at a hotel and rented a car for 4 days. If Connor paid $675, it means that

3x + 4y = 675 - - - - - - - - - - - 1

Jillian stayed at the same hotel, but spent 4 nights and rented a car for 5 days from the same company. If Jillian paid $875, It means that

4x + 5y = 875 - - - - - - - - - - - -2

Multiplying equation 1 by 4 and equation 2 by 3, it becomes

12x + 16y = 2700

12x + 15y = 2625

Subtracting, it becomes

y = 75

Substituting y = 75 into equation 1, it becomes

3x + 4 × 75 = 675

3x + 300 = 675

3x = 675 - 300 = 375

x = 375/3 = $125

Answer:

A. three planes intersecting at a point

Step-by-step explanation:

Whenever three planes intersect at a point, then we have an ordered triplet (x,y,z) that is a solution to the associated system of equation.

This is a unique solution and it is finite.

However three planes intersecting at a line gives infinitely many solutions.

Three or two parallel planes may have no solution or or infinitely many solutions when they coincide.

Answer:

x= 7.5 mile/h

y= 3.5 miles / h

Step-by-step explanation:

Given x and y

so, x+y = speed down river

and x-y= speed up river

using

travel time = distance / speed for each case;

we get;

22/x+y=2  ---(1)

and

22/x-y=5.5 -----(2)

Solving equations simultaneously

eq 1 gives

2x+2y=22  ---(3)

eq 2 gives

5.5x-5.5y=22 ----(4)

Multiply eq 3 by 2.75

==> 5.5x+5.5y=60.5 --- (5)

adding eq 4 with eq 5

==> 11x =82.5

==> x= 7.5 miles/h

put this value in eq 3

==> 2(7.5)+2y=22

==> 2y=22-19.8

==> 2y=7

==> y = 3.5

Once setting up two equations based on downstream and upstream velocities, using variables x for Pratap's rowing speed and y for the current's speed, you can solve for these variables. The results show Pratap can row at a speed of 7.5 mph in still water, while the rate of the current is 3.5 mph.

Firstly, we have to understand that Pratap's speed with the current and against the current will be affected. Let's say x is the speed of Pratap rowing in still water, and y is the speed of the current. When going downstream (in the direction of the current), Pratap's speed is (x + y) since the current helps him. He covered a distance of 22 miles in 2 hours, so the rate (velocity) can be found by dividing distance by time. Hence, x + y = 22/2 = 11 miles per hour.

On the other hand, when rowing upstream (against the current), Pratap's speed is (x - y), because now the current hinders his rate. He covered the same distance in 5 1/2 hours, so x - y = 22/(5.5) = 4 miles per hour.

Now, we have two equations: x + y = 11 and x - y = 4. Solving these equations (by adding them), we find that 2x = 15 hence x = 7.5. Pratap can row in still water at a speed of 7.5 miles per hour. Substituting x=7.5 into either of the original equations we get y = 3.5. Hence, the rate of the current is 3.5 miles per hour.

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Answer:

4r

Step-by-step explanation:

If you take away the variable, add the 3 and metaphorical 1

3 + 1 = 4

Adding like terms (3+1) and then putting the variable back in makes 4r

Hope this helps, mark brainliest

Answer:

4r

Step-by-step explanation:

I hope this helps!

Answer: [tex]H'(4,-3)[/tex]

Step-by-step explanation:

For this exercise you must remember that the original figure (before a transformation) is called "Pre-Image" and the one obtained after the transformation is called "Image".

 A Translation is defined as a transformation in which the figure is moved a  fixed distance in a fixed direction. In this kind of transformatiosn the size and shape do not change and the figure is not flipped.

In this case you know that the Pre-Image is the Triangle FGH.

You can identify in the picture that its vertex H has these coordinates:

[tex]H(1,-2)[/tex]

Where:

[tex]x=1\\y=-2[/tex]

Since the rule is:

[tex](x,y)[/tex] → [tex](x+3,y-1)[/tex]

You can substitute the coordinates of H into the given rule in order to find the coordinates of H'. This is:

[tex]H'=(1+3,-2-1)=(4,-3)[/tex]

The coordinates of H after translation using the rule (x,y) > (x+3,y-1) are obtained by adding 3 to the x-coordinate and subtracting 1 from the y-coordinate of H's original position.

To determine the coordinates of H after applying the translation rule (x,y) > (x+3,y-1), you have to add 3 to the x-coordinate and subtract 1 from the y-coordinate of point H's original position.

For example, if H originally has coordinates (a,b), then after the translation, the new coordinates of H will be (a+3, b-1).

The given translation rule does not change regardless of the element being translated, be it point or vector, as the rule is applied to each coordinate independently.

Therefore, if you know the original coordinates of point H, you just apply the rule directly to find the translated coordinates.

Answer:

Length of bases of the trapezoid are 5 inch and 15 inch.

Step-by-step explanation:

Let the length of the second base be 'x'.

Given:

one base of a trapezoid is 3 times the length of the second base.

So we can say that;

Length of the first base = [tex]3x[/tex]

Also Given:

the height of the trapezoid is 2 in smaller than the second base

So we can say that;

Height of the trapezoid = [tex]x-2[/tex]

Area of the trapezoid = 30

We need to find the length of both the bases.

Solution:

Now we know that;

Area of trapezoid is equal to half of the sum of the length of two bases multiplied by height of the trapezoid.

framing in equation form we get;

[tex]\frac{x+3x}{2}\times(x-2)=30\\\\\frac{4x}{2}\times (x-2)=30\\\\2x(x-2)=30[/tex]

Now Dividing both side by 2 we get;

[tex]\frac{2x}{2}(x-2)=\frac{30}{2}\\\\x(x-2)=15\\\\x^2-2x-15=0[/tex]

Now we will find the roots by factorizing the equation we get;

[tex]x^2-5x+3x-15=0\\\\x(x-5)+3(x-5)=0\\\\(x+3)(x-5)=0[/tex]

Now we find the values of x by solving separately we get;

[tex]x+3=0 \ \ \ \ Or \ \ \ \ \ \ x-5=0\\\\x=-3 \ \ \ \ \ \ \ Or \ \ \ \ \ \ x=5[/tex]

Now we have got 2 values of 'x' one positive and one negative.

Now we know that length of the base of trapezoid cannot be negative hence we will discard it and consider the positive value of 'x'.

Length of second base = 5 in

Length of first base = [tex]3x=3\times5 =15\ in[/tex]

Hence Length of base of the trapezoid are 5 inch and 15 inch.

Using the Pythagorean theorem, the distance from the bottom of the ladder to the wall is found to be 5 ft, after solving the quadratic equation that arises from the conditions given.

The question asks for the distance from the bottom of the ladder to the wall when a ladder is leaning against it. We can use the Pythagorean theorem to solve this problem as it forms a right-angled triangle. Let's denote the distance from the bottom of the ladder to the wall as x, then the distance from the top of the ladder to the bottom of the wall would be x + 7 ft. Since the ladder's length, which represents the hypotenuse, is 13 ft, we can set up the equation:

x2 + (x + 7)2 = 132

Expanding this, we get:

x2 + x2 + 14x + 49 = 169

Combining like terms:

2x2 + 14x - 120 = 0

Dividing everything by 2 to simplify:

x2 + 7x - 60 = 0

Factoring the quadratic equation:

(x + 12)(x - 5) = 0

The possible values for x are -12 and 5. Since distance cannot be negative, the distance from the bottom of the ladder to the wall is 5 ft.

Answer:

Step-by-step explanation:

Variability refers to the degree in which a set of data set spreads out or dispersed or clustered together. There are four measure of variability namely: range, interquartile range (IQR) variance and standard deviation. The consequence of increasing variability is that the distance from one score to another tends to increase, and a single score tends to provide a less accurate representation of the entire distribution.

In the context of a data set, increasing variability implies that scores in the dataset are dispersing more, increasing the distance from one score to another. Therefore, a single score provides a less accurate reflection of the entire distribution. The correct answer to the question is B.

The subject of the question relates to the concept of variability in a data set, which is a measure of how much scores differ from each other in a distribution. The question asks about the consequences of increasing variability.

If variability increases, it means that the scores in the dataset are spreading out more. In other words, the distance from one score to another tends to increase. Consequently, a single score tends to provide a less accurate representation of the entire distribution because the scores are more spread out and there is a higher level of diversity in the scores. So, the correct answer is B.

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Answer:

No

Step-by-step explanation:

The majority fairness criteria states that a person with majority of votes should be the winner and it is not violated by the plurarity method. Rather pluarity is always satisfied with majority fairness criterion.

Answer:

The Spanish ship goes to Port Said and the French ship carries tea. However, tea can be carried by the Brazilian ship, too.

Step-by-step explanation:

French 5:00 tea blue Genoa

Greek 6:00 coffee red Hamburg

Brazilian 8:00 cocoa black Manila

English 9:00 rice white Marseille

Spanish 7:00 corn green Port Said

Answer:

i) amplitude is 4/2 =2

ii) period of the graph  = 3.14 ( radians), (or 180°)

iii) the frequency of the given function is f  = 2Hz ( hertz).

iv) The midline for the function is y = -3.

v) The maximum value of the problem of the function is -1.

vi) the minimum value of the function is -5.

Step-by-step explanation:

i) the Peak to Peak is = -1 - (-5) = 4

  therefore amplitude is 4/2 =2

ii) The period of the graph  = 3.14

iii) one cycle is over 2π radians

    one cycle of the given function is over π radians. Therefore the given    

     function will have 2 cycles over 2π radians. Therefore the frequency of the given function is f  = 2 hertz.

iv) The midline for the function is y = -3.

v) The maximum value of the problem of the function is -1

vi) the minimum value of the function is -5.

The value of a is 80

Step-by-step explanation:

The distance of a point [tex](x_0,y_0)[/tex] from the y-axis can be written as

[tex]d_y = |x_0|[/tex]

because the x-coordinate of the y-axis is zero.

Similarly, the distance of a point [tex](x_0,y_0)[/tex] from the x-axis can be written as

[tex]d_x=|y_0|[/tex]

Since the y-coordinate of the x-axis is zero.

In this problem:

- The distance of the point A (−30, −45) from the y-axis can be written as

[tex]d_A = |-30|=30[/tex]

- The distance of point B (a,a) from the x-axis can be written as

[tex]d_B = |a| = a[/tex]

Since [tex]a>0[/tex].

We are told that 2/3 of the distance from the y-axis to point A (−30, −45) is equal to 1/4 of the distance from the x-axis to point B(a, a), which means

[tex]\frac{2}{3}d_A = \frac{1}{4}d_B[/tex]

Therefore,

[tex]\frac{2}{3}(30)=\frac{1}{4}a[/tex]

And solving for a,

[tex]20=\frac{1}{4}a\\a=80[/tex]

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Answer:

40

Step-by-step explanation:

Answer:D

Step-by-step explanation:

The solution is: : $117,783.05 will be her balloon payment at the end of 7 years if she chooses this mortgage.

Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.

here, we have,

Step 1

Calculate the number of monthly payments for 7 years.

The formula we would be using is given as:

Monthly payment = [Loan amount × (rate/number of year)] ÷ [1 - (1 +r/n)^nt)]

Loan amount =$146,000

Rate = 4.75% = 0.0475

n = number of payments = 12

t= number of years = 23

Monthly payment = [ 146,000× (0.0475/12)] ÷ [1 - (1 +0.0475/12)^-276)]

Monthly payment = $870.49

Step 2

Calculate the amount left after 7 years using the formula

Ballon payment after 7 years = Loan amount ( 1 + r)ⁿ - P[(1 +r)ⁿ - 1 /r]

r = 4.75% for 12 years

= 146,000( 1 + 0.0475/12)^84 - 870.49[((1 + 0.0475/12)^84 - 1)/0.0475/12]

= $117,783.05

Hence, The solution is: : $117,783.05 will be her balloon payment at the end of 7 years if she chooses this mortgage.

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complete question:

Yvette is considering a 7/23 balloon mortgage with an interest rate of 4.75%

to purchase a house for $146,000. What will be her balloon payment at the end of

7 years if she chooses this mortgage?

Answer:

5x + 13

Step-by-step explanation:

(3x + 9 ) + (2x + 4)

It's quite simple, we just have to add the parts that have x to each other and those that don't have x to each other

3x + 2x + 9 +4 =

then the result is

5x + 13

Answer:

The answer to your question is the last choice

Step-by-step explanation:

                               [tex](4z^{8})^{-3}[/tex]

Process

1.- Use the exponent law "Power"

                               [tex]4^{-3} z^{(8)(-3)}[/tex]

2.- Simplify

                               [tex]4^{-3} z^{-24}[/tex]

3.- Use the exponent law "Negative exponent"

                               [tex]\frac{1}{4^{3} z^{24}}[/tex]

                               [tex]\frac{1}{64z^{24}}[/tex]

4.- Conclusion    

He should have applied the exponent -3 to 4 to get   [tex]\frac{1}{64z^{24}}[/tex]            

Answer:

d

Step-by-step explanation:

Answer:

.

Step-by-step explanation:

a.) 72 ÷ 8 × 9 = 81 (we divide 72 by 8 first then multiply the result with 9)

b.) -72 ÷ 8 × 9 = -81 (it's same with a only differ by negative sign)

c.) 72 ÷ (-8) × 9 = -81 (dividing 72 by -8 will give us -9 and multiplying -9 by -9 will give the result of -81)

d.) 72 ÷ 8 × (-9) = -81 (divide 72 by 8 and it will be 9, mutliply it by -9 and again it will give -81)

e.) -72 ÷ 8 × (-9) = 81 (divide -72 by 8 and it will be -9 multiplying it by -9 will give a positive 81 since two negative signed numbers multiplied or divided gives positive result)

Answer:

A)81

B)-81

C)-81

D)-81

E)81

Step-by-step explanation:

Answer: the maximum length of the sandbox is 42 feet.

the maximum width of the sandbox is 16 feet.

Step-by-step explanation:

Let L represent the length of the sandbox.

Let W represent the width of the sandbox.

The formula for determining the perimeter of a rectangle is expressed as

Perimeter = 2(L + W)

The perimeter of the sandbox must be less than or equal to 116 feet. This means that

2(L + W) ≤ 116

Dividing through by 2, it becomes

L + W ≤ 116/2

L + W ≤ 58 - - - - - - - - -1

The length of the sandbox will be 6 less than 3 times the width. This means that

L = 3W - 6

Substituting L = 3W - 6 into equation 1, it becomes

3W - 6 + W ≤ 58

4W ≤ 58 + 6

4W ≤ 64

W ≤ 64/4

W ≤ 16

L = 3W - 6 = 3 × 16 - 6

L ≤ 42

Answer:

Therefore,

[tex]m\angle R = 69.4\°[/tex]

Step-by-step explanation:

Given:

In Right Angle Triangle PQR at ∠Q = 90° such that

RQ = 3     ....Adjacent side of angle R

PQ = 8     ....Opposite side of angle R

To Find:

m∠R = ?

Solution:

In Right Angle Triangle PQR, Tan Identity,

[tex]\tan R= \dfrac{\textrm{side opposite to angle R}}{\textrm{side adjacent to angle R}}[/tex]

Substituting the values we get

[tex]\tan R= \dfrac{PQ}{QR}=\dfrac{8}{3}=2.6666\\\\\angle R=\tan^{-1}(2.666)=69.439=69.4\°[/tex]

Therefore,

[tex]m\angle R = 69.4\°[/tex]

Answer:

summation of five times negative three to the power of n from n equals zero to infinity

Step-by-step explanation:

Summation Notation

It represents the sum of a finite or infinite number of terms. Let's analyze the terms of the given succession:

5-15+45-135+...

If we take 5 as a common factor, we have

5(1-3+9-27+...)

The parentheses contain the alternate sum/subtraction of powers of 3. The odd terms are positive, the even terms are negative, thus the exponent must be n starting from 0 or n-1 starting from 1

The summation is then represented by

[tex]\sum_{n=0}^{\infty}5(-3)^n[/tex]

This corresponds with the option:

summation of five times negative three to the power of n from n equals zero to infinity

Final answer:

The correct summation notation for the series 5 - 15 + 45 - 135 + ... is the summation of five times negative three to the power of n from n equals zero to infinity.

Explanation:

The given sequence is 5 - 15 + 45 - 135 + ... which can be seen as a geometric series with a pattern of alternating signs and a common ratio of -3. The first term of the sequence is 5 (when n=0), and each subsequent term is multiplied by -3. Therefore, to write this pattern using summation notation, the nth term can be represented as 5 × (-3)^n. So, the summation notation for the entirety of the series from n equals 0 to infinity is:

Σ [5 × (-3)^n], from n = 0 to infinity

This matches the option: summation of five times negative three to the power of n from n equals zero to infinity.

Final answer:

After calculating the z-scores for a high temperature of 55 degrees in both January and July, it was found that a temperature of 55 degrees is more unusual in July than in January due to the higher absolute value of its z-score.

Explanation:

To determine in which month it is more unusual to have a day with a high temperature of 55 degrees, we compare the standardized scores (z-scores) of 55 degrees for January and July. The z-score is calculated by subtracting the mean from the observation and then dividing the result by the standard deviation. For January, with a mean of 36 and standard deviation of 10, the z-score is (55 - 36) / 10 = 1.9. For July, with a mean of 74 and standard deviation of 8, the z-score is (55 - 74) / 8 = -2.375.

Comparing the absolute values of the z-scores, the z-score for July is higher in absolute value, indicating that a temperature of 55 degrees is more unusual in July than it is in January.

Answer:

The container can hold 24 gallons of water.

Step-by-step explanation:

Let the Amount of water in gallons container can hold be 'x'.

Now given:

the container that holds the water for the football team is  1/4 full.

therefore Amount of water in container = [tex]\frac14x[/tex]

We need to find the Amount of water in gallons container can hold.

Solution:

Now given:

after pouring in 10 gallons of water, it is 2/3 full.

So equation can be framed as;

[tex]\frac14x+10=\frac23x[/tex]

Now Combining the like terms we get;

[tex]10=\frac23x-\frac14x[/tex]

Now We will use LCM to make the denominator common we get;

[tex]10=\frac{2x\times4}{3\times4}-\frac{1x\times3}{4\times3}\\\\10=\frac{8x}{12}-\frac{3x}{12}[/tex]

Now the denominators are common so we will solve the numerator we get;

[tex]10=\frac{8x-3x}{12}\\\\10=\frac{5x}{12}[/tex]

Now Multiplying both side by [tex]\frac{8}{12}[/tex] we get;

[tex]10\times\frac{12}{5}=\frac{5x}{12}\times \frac{12}{5}\\\\x=24 \ gallons[/tex]

Hence The container can hold 24 gallons of water.

Answer:b also d

Step-by-step explanation:

Adrian's recipe has a greater ratio of sauce to dough, and the ratios can be compared by comparing 15 to 18 because 30 is in both dough columns.

In Adrian's recipe, the ratio of sauce to dough is 15:30, which simplifies to 1:2.

In Juliet's recipe, the ratio of sauce to dough is 18:30, which also simplifies to 1:2.

Since both ratios simplify to the same value of 1:2, it means that Adrian's and Juliet's recipes have equal ratios of sauce to dough.

Therefore, the statement "Adrian's and Juliet's recipes have equal ratios of sauce to dough" is true.

The reason we can compare the ratios by looking at 15 and 18 is because both of these values represent the amount of sauce, and 30 represents the amount of dough in both recipes.

By comparing the sauce portions directly, we can determine that the ratios are indeed equal.

It's important to note that the absolute values of the sauce and dough quantities are less relevant in this context than their proportions, which are expressed in the ratios.

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Answer:

4.8 hours

Step-by-step explanation:

Distance = speed × time

If Tim drove 4 hours at 60 miles per hour

Than he drove a total of 4×60 miles

He drove 240 miles

Time = distance ÷ speed

If he drove 240 miles at 50 miles per hour

Than he took 240÷50 hours

He took 4.8 hours (4 hours and 48 minutes)

Final answer:

To find the intersection points of the paths of two particles, one needs to set their position functions equal and solve for the variable t. If no solution exists, then the intersection points do not exist, and the answer is DNE.

Explanation:

To find the points where the paths of two particles intersect, we set their respective position functions r1(t) and r2(t) equal to each other and solve for t. The position functions given are:
r1(t) = (t, t^2, t^3)
r2(t) = (1 + 4t, 1 + 16t, 1 + 52t).

Their paths intersect when:
x1 = x2
y1 = y2
z1 = z2.

By solving these equations, we find the common t values that make both position vectors equal. If there is no common t that satisfies all three conditions, then the particles never collide, and the answer is DNE (Does Not Exist).