Please help me on this question​

The derivative exists at x=0 and its value is 0. So, f'(0) = 0.

The derivative of a function can be thought of as the slope of a tangent line to the function's graph at a specific point. This tangent line represents the instantaneous rate of change of the function at that point. The derivative of a function can be used to find critical points, which are points where the function reaches a maximum or minimum value, and to find the concavity of the function, which indicates the direction in which the function is changing.

The derivative can be calculated using different methods, including the limit definition of the derivative, the power rule, the product rule, the quotient rule, and the chain rule. The derivative can also be used to calculate the slope of a curve, the velocity of a moving object, and the acceleration of an object.

To find f'(0) using the definition of a derivative, we need to use the following formula:

f'(a) = lim(h->0) [f(a+h) - f(a)]/h

where "a" is the point at which we want to find the derivative, and "h" is a small value that approaches zero.

In this case, we want to find f'(0). Let's use the formula and simplify the expression:

f'(0) = lim(h->0) [f(0+h) - f(0)]/h

= lim(h->0) [h²sin(1/h)]/h (since f(0)=0)

= lim(h->0) [hsin(1/h)]

Now, we need to evaluate this limit. We can rewrite the limit using the squeeze theorem, which says that if we have two functions, g(x) and h(x), such that g(x) ≤ f(x) ≤ h(x) for all x in some interval, and if lim(x->a) g(x) = lim(x->a) h(x) = L, then lim(x->a) f(x) = L.

In our case, we have:

-1 ≤ sin(1/h) ≤ 1 for all h≠0 (using the fact that -1 ≤ sin(x) ≤ 1 for all x)

-h ≤ hsin(1/h) ≤ h for all h≠0 (multiplying by h, which is always non-negative)

Therefore, by the squeeze theorem, we have:

lim(h->0) -h = 0 = lim(h->0) h

Since both limits are equal, we can conclude that the limit of hsin(1/h) as h approaches 0 exists and is equal to 0.

Thus, f'(0) = 0.

Therefore, the derivative exists at x=0 and its value is 0. So, f'(0) = 0.

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A) To find the annual rate of change, we can use the formula:

r = (V2/V1)^(1/n) - 1

where:

V1 = initial value ($42,000)

V2 = final value ($11,000)

n = number of years (2006 - 1994 = 12)

Plugging in the values, we get:

r = (11000/42000)^(1/12) - 1 ≈ -0.1135

Therefore, the annual rate of change between 1994 and 2006 is approximately -0.1135.

B) To convert the rate of change to a percentage, we can multiply by 100 and add a percent sign:

r = -0.1135 × 100% ≈ -11.35%

Therefore, the correct answer to part A written in percentage form is approximately -11.35%.

C) Assuming the car value continues to drop by the same percentage, we can use the formula for exponential decay:

V = V0 * (1 - r)^t

where:

V0 = initial value ($11,000 in 2006)

r = annual rate of change (-0.1135)

t = number of years (2010 - 2006 = 4)

Plugging in the values, we get:

V = 11000 * (1 - (-0.1135))^4 ≈ $6,250

Therefore, the value of the car in the year 2010 would be approximately $6,250, rounded to the nearest 50 dollars.

Using the formula σr = 3Fx/yz², we can easily compute the Modulus of Rupture.

The term "bending strength" is occasionally used to refer to the measure of a specimen's strength prior to rupture, also known as the "modulus of rupture," or MOR.

Contrary to the modulus of elasticity, which measures the wood's deflection but not its total strength, it can be used to evaluate a species' strength.

The formula σr = 3Fx/yz² for the load force F and the material's size dimensions in the three directions of x, y, and z can be used to compute the modulus of rupture, or "sigma."

The external force applied to the substance of interest in this instance is the load.

Therefore, using the formula σr = 3Fx/yz², we can easily compute the Modulus of Rupture.

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Correct question:

How do you compute MOR?

The length of major arc PQ is 35π units.

What is major arc?

In geometry, an arc is a portion of the circumference of a circle. A major arc is an arc that spans more than 180 degrees of the circle. In other words, a major arc is an arc that is greater than a semicircle (which spans exactly 180 degrees).

we know that,

length of arc = s = 2 π r (θ/360°)

where, r is the radius and θ is the angle of arc.

we have, r = 30

               θ = 210

so, length of major arc PQ :

   =  2 π r (θ/360°)

   = 2 π × 30 × (210/360)

   = 60  π  × 7/12

   = 35π units.

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The tourist has to pay the government $3,825,000.

make 45% into a decimal (0.45) and then multiply it by 8,500,000.

The determinant of the matrix A is 70.

The determinant of a 3x3 matrix can be found using the following formula:

|A| = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

where aij represents the element in the ith row and jth column of the matrix.

Using this formula, we can find the determinant of the given matrix:

|A| = 2(82 - 4(-2)) - 3(-12 - 4(-6)) + 0(-1*(-2) - 8*(-6))

= 16 + 54 + 0

= 70

Therefore, the determinant of the matrix A is 70.

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Answer:

[tex]a=3\sqrt{5}[/tex] or 6.7 approximately (Yes you did it correctly)

Step-by-step explanation:

Find leg [tex]a[/tex] of a right triangle if leg [tex]b[/tex] =6 and hypotenuse = 9.

To find side [tex]a[/tex] use Pythagorean Theorem:

[tex]a^2+b^2=c^2[/tex]

After substituting b = 6 and c = 9 we have:

[tex]a^2+6^2=9^2[/tex]

[tex]a^2=9^2-6^2[/tex]

[tex]a^2=81-36[/tex]

[tex]a^2=45[/tex]

[tex]a=\sqrt{45}[/tex]

[tex]a=3\sqrt{5}[/tex]

[tex]a=6.7[/tex]

Julia can expect to roll a 3 approximately 61 times if she rolls the die 200 times.

Probability is a means to gauge the likelihood that an event will take place. It is symbolized by the numbers 0 and 1, respectively, with 0 denoting an improbable event and 1 denoting an inescapable event. With two equally likely outcomes, switching a valid coin and coin flips have a chance of 0.5, or 50%. head or tail. A subfield of mathematics called probability theory examines random events rather than their attributes. It is employed in many different fields, including finance, science, building, and facts and statistics.

we found the best estimate for the probability of rolling a 3 to be:

p = (30 + 0.5) / 100 = 0.305

This means that the probability of rolling a 3 on any one roll of the die is 0.305.

If Julia rolls the die 200 times, we can use the formula for the expected value of a binomial distribution to find the expected number of 3's she will roll:

E(X) = np

where:

n = number of trials = 200

p = probability of success (rolling a 3) = 0.305

So, plugging in the values:

E(X) = 200 × 0.305

E(X) = 61

Therefore, Julia can expect to roll a 3 approximately 61 times if she rolls the die 200 times.

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Answer:

Michael needs to purchase 6 1/4 feet of metal flashing.

Step-by-step explanation:

To determine the number of feet of metal flashing that Michael needs to purchase, we need to add the lengths of the two small pieces he already has:

3 1/8 feet + 1 3/4 feet

To add these two fractions, we need to have a common denominator. In this case, the smallest common denominator is 8:

3 4/8 feet + 1 14/8 feet = 4 18/8 feet

Now we can simplify the mixed number by converting it to an improper fraction:

4 18/8 feet = (4 x 8 + 18)/8 feet = 50/8 feet

So Michael already has 50/8 feet of metal flashing. To determine how much more he needs to purchase, we need to subtract this amount from the total amount he needs:

6 1/2 feet - 50/8 feet

To subtract these two fractions, we again need to have a common denominator. In this case, the smallest common denominator is 8:

6 4/8 feet - 50/8 feet = (6 x 8 + 4)/8 feet - 50/8 feet = 52/8 feet - 50/8 feet = 2/8 feet

So Michael needs to purchase an additional 2/8 feet of metal flashing to complete his household project, which simplifies to 1/4 feet. Therefore, Michael needs to purchase a total of:

50/8 feet + 1/4 feet = 6 1/4 feet

Answer:

12/20 = 0.6

Step-by-step explanation:

The total number of cards in Kayla's hand is:

6 (spades) + 7 (hearts) + 5 (diamonds) + 2 (clubs) = 20

The number of red cards (hearts or diamonds) in Kayla's hand is:

7 (hearts) + 5 (diamonds) = 12

Therefore, the probability of randomly selecting a red card from Kayla's hand is:

12/20 = 0.6

Answer: 0.6

Step-by-step explanation: you add the total amount of cards she has which is 20(6+7+5+2) then the amount of hearts or diamonds is 7+5 for hearts and diamonds, which is =12 so the amount of hearts and diamonds divided by the total amount of cards, 12/20 which is 6/10 when divided by 2, 6/10 is 0.6, if you want to go further 6/10 can be broken down to 3/5 which is also 0.6

Answer:

[tex] \frac{ {6}^{5} }{ {6}^{ - 2} } = {6}^{5 - ( - 2))} = {6}^{5 + 2} = {6}^{7} [/tex]

The answer of the given question based on the matching  the equation with the graph is , Y=3x -5 , Y= -5x + 3  , Y= x - 1  , Y= (x + 3)² -2  , Y= 1/2x + 2 , Y= x .

An equation is  mathematical statement that shows that two expressions are equal. It consists of two sides, left-hand side (LHS) and right-hand side (RHS), separated by equal sign (=). The LHS and RHS may contain numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. The purpose of an equation is to solve for the value of an unknown variable.

Equations are used in many areas of mathematics and science, including algebra, calculus, physics, and engineering. They are essential for modeling and solving real-world problems and are a fundamental tool for understanding the natural world.

Here are the equations that match the graphs in the given image:

Y=3x -5 (Line passing through points (0,-5), (1,-2), and (2,1))

Y= -5x + 3 (Line passing through points (0,3) and (1,-2))

Y= x - 1 (Line passing through points (0,-1) and (1,0))

Y= (x + 3)² -2 (Parabola opening upwards with vertex at (-3,-2))

Y= 1/2x + 2 (Line passing through points (0,2) and (2,3))

Y= x (Line passing through points (-2,-2), (0,0), and (2,2))

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Answer:

Let's start by assigning variables to represent the number of nickels, dimes, and quarters in the jar.

Let x be the number of nickels.

Then the number of dimes is 3 times as many as nickels, so the number of dimes is 3x.

And the number of quarters is twice as many as dimes, so the number of quarters is 2(3x) = 6x.

We know that the total amount of money in the jar is $5.55, which is equal to:

0.05x (for the value of the nickels) + 0.10(3x) (for the value of the dimes) + 0.25(6x) (for the value of the quarters)

Simplifying this expression, we get:

0.05x + 0.30x + 1.50x = 5.55

Combining like terms, we have:

1.85x = 5.55

Dividing both sides by 1.85, we get:

x = 3

So there are 3 nickels in the jar.

Using this value, we can find the number of dimes and quarters:

Number of dimes = 3x = 3(3) = 9

Number of quarters = 6x = 6(3) = 18

Therefore, there are 3 nickels, 9 dimes, and 18 quarters in the jar.

Step-by-step explanation:

Answer: A

Step-by-step explanation:

The answer is A.

The hypotenuse of the right angled triangle is [tex]$14\sqrt{2}$[/tex].

What is hypotenuse?

The hypotenuse is the longest side of a right-angled triangle and is opposite to the right angle. It is the side that is opposite to the 90-degree angle and is located opposite the right angle.

To solve this problem, we can use the concept of special right triangles, specifically the 45-45-90 right triangle. In a 45-45-90 right triangle, the sides are in the ratio of 1:1:sqrt(2).

Let's label the length of the triangle as [tex]$L$[/tex], the hypotenuse as [tex]$H$[/tex], and the angle between the hypotenuse and base length as [tex]45[/tex]°.

We are given that the angle formed between the hypotenuse and the base length is 45 degrees. In a 45-45-90 right triangle, the two legs are congruent, so the ratio of the sides is 1:1:sqrt(2).

Since we know that the ratio of the sides is 1:1:sqrt(2), we can set up the following equation:

[tex]$L:L:H = 1:1:\sqrt{2}$[/tex]

We are given that [tex]$L = 14$[/tex], so we can substitute this value into the equation:

[tex]$14:14:H = 1:1:\sqrt{2}$[/tex]

Solve for H.

Since the ratio of the sides is 1:1:sqrt(2), we know that [tex]$H$[/tex] is equal to [tex]$L$[/tex] multiplied by [tex]$\sqrt{2}$[/tex]. Therefore, we can solve for [tex]$H$[/tex] by multiplying [tex]$14$[/tex] by [tex]$\sqrt{2}$[/tex]:

[tex]$H = 14\sqrt{2}$[/tex]

Therefore, the hypotenuse of the right angled triangle is [tex]$14\sqrt{2}$[/tex].

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Answer:

[tex]384 \: {cm}^{3} [/tex]

Step-by-step explanation:

Given:

h = 8 cm

a (base) = 48 cm^2

Find: V - ?

[tex]v = a(base) \times h[/tex]

[tex]v = 48 \times 8 = 384 \: {cm}^{3} [/tex]

as already suggested, we can simply get the whole volume of the larger cone and then get the volume of the upper-smaller cone and if we subtract the volume of the upper-smaller cone, in essence making a hole in the larger cone, what's leftover is the shaded part.

[tex]\stackrel{ \textit{\LARGE Larger} }{\textit{volume of a cone}}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=4\\ h=10 \end{cases}\implies V=\cfrac{\pi (4)^2 (10)}{3} \\\\\\ \stackrel{ \textit{\LARGE Upper-Smaller} }{\textit{volume of a cone}}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=2\\ h=4 \end{cases}\implies V=\cfrac{\pi (2)^2 (4)}{3} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\cfrac{\pi (4)^2 (10)}{3}~~ - ~~\cfrac{\pi (2)^2 (4)}{3}\implies \cfrac{160\pi }{3}-\cfrac{16\pi }{3}\implies \cfrac{160\pi -16\pi }{3} \\\\\\ \cfrac{144\pi }{3} ~~ \approx ~~ \text{\LARGE 150.80}~cm^2[/tex]

Based on the information provided, the visibility distance for Pretoria in kilometers is 19.30 kilometers.

The table shows the visibility from Pretoria is 12, miles, while the visibility in Cape Town is 6 miles. This implies we need to change miles to kilometers. These two units are used for distance but they are not equivalent as 1 mile =  1.609 kilometers, knowing this, let's calculate the distance in kilometers:

12 miles x 1.609 =  19.30 kilometers.

Note: This question is incomplete, below I attach the missing information:

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Answer:

The quotient is Q(x) = -2x² + x + 6 and the remainder is R(x) = 52.

Step-by-step explanation:

           -2x² + x + 6

    ------------------------

x - 7 | -2x³ + 15x² - x + 10

     -(-2x³ + 14x²)

     ---------------

              x² - x

              -(x² - 7x)

              ---------

                  6x + 10

                  -(6x - 42)

                  ----------

                          52

The correct answer is that if someone simply gets a high school certificate, their salary is most likely to make less than $40,000.

In general, the phrase "income" refers to the sum of money, assets, and other transfers that are worth something obtained over a predetermined period period of time as an exchange for goods or services.

Only having completed high school:

51>24>3

Therefore, it is highly likely that they make less than $40,000.

Has a salary of at least 60,000.

6<17<48

Therefore, a Master's degree is most likely their greatest level of education.

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Answer:

6.1 cm

Step-by-step explanation:

We have to find the hypotenuse of the orange triangle which will be the leg of the green triangle

[tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex]

[tex]4.2^{2}[/tex] + [tex]4^{2}[/tex] = [tex]c^{2}[/tex]

17.64 + 16 = [tex]c^{2}[/tex]

33.64 = [tex]c^{2}[/tex]

[tex]\sqrt{33.64}[/tex] = [tex]c^{2}[/tex]

5.8 = c

[tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex]

[tex]2^{2}[/tex] + [tex]5.8^{2}[/tex] = [tex]c^{2}[/tex]

4 + 33.64 = [tex]c^{2}[/tex]

37.64 = [tex]c^{2}[/tex]

[tex]\sqrt{37.64}[/tex] = [tex]\sqrt{c^{2} }[/tex]

6.13514466007 = c  This is a rounded number.  This is an irrational number.  It never repeats or terminates.

Rounded to 1 decimal place

6.1

Helping in the name of Jesus.

Answer:

B

Step-by-step explanation:

Answer and Explanation: Nuclear fusion takes place at very high temperature and pressure so that the two elements fuse together, forming new, heavier elements. The nuclear fusion reaction releases a massive amount of energy, and it normally takes place in the core of the stars.

hope that was the answer

Answer:

Its c it has to be because its merge and if u look it up it shows and that's what it means yw!

Step-by-step explanation:

Answer:

If the recipe calls for 1 1/4 pounds of beets and Sandra removes 1/2 of a pound from the scale, then the amount of beets she has left is:

1 1/4 - 1/2 = 1 - 1/4 - 1/2 = 3/4 pounds

Therefore, Sandra has 3/4 pounds of beets, which is exactly what she needs for the chocolate beet cake recipe

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Answer:

Step-by-step explanation:

To find the mean absolute deviation (MAD), we first need to find the mean of the data set:

mean = (2 + 5 + 3 + 5 + 4 + 3 + 3 + 5 + 4 + 4) / 10

mean = 4

Next, we find the absolute deviation of each data point from the mean:

|2 - 4| = 2

|5 - 4| = 1

|3 - 4| = 1

|5 - 4| = 1

|4 - 4| = 0

|3 - 4| = 1

|3 - 4| = 1

|5 - 4| = 1

|4 - 4| = 0

|4 - 4| = 0

Then, we find the mean of these absolute deviations:

MAD = (2 + 1 + 1 + 1 + 0 + 1 + 1 + 1 + 0 + 0) / 10

MAD = 0.8

Therefore, the mean absolute deviation of the data set is 0.8.

Answer: 54.43 miligrams

Step-by-step explanation:

To calculate the dose of ibuprofen for an 18-month-old child weighing 24 pounds, we need to convert the weight from pounds to kilograms.

1 pound is equal to 0.453592 kilograms.

So, the weight of the child in kilograms is:

24 pounds × 0.453592 kg/pound = 10.886kg (rounded to three decimal places)

Now we can use the given dosage information to calculate the usual dose of ibuprofen for the child.

The usual dose of ibuprofen is 5 mg/kg of body weight when the fever is under 102.5 degrees Fahrenheit.

So, for the 18-month-old child weighing 10.886 kg, the usual dose of ibuprofen would be:

5 mg/kg × 10.886 kg = 54.43 mg

Therefore, the usual dose of ibuprofen for an 18-month-old child weighing 24 pounds is 54.43 mg.

Answer:

First, we need to convert the weight of the child from pounds to kilograms since the dosage is given in milligrams per kilogram.

We can use the conversion factor: 1 pound = 0.453592 kilograms.

So, 24 pounds = 24 x 0.453592 = 10.8862 kg (rounded to 4 decimal places).

Next, we can calculate the usual dose of Ibuprofen by multiplying the weight of the child (in kg) by the dose per kg:

Usual dose = 5 mg/kg x 10.8862 kg = 54.431 mg

Therefore, the usual dose of Ibuprofen for an 18 month old weighing 24 pounds would be 54.431 mg when the fever is under 102.5 degrees Fahrenheit. However, it is important to note that dosages may vary depending on the specific circumstances, and it is always best to consult a healthcare provider for proper dosing instructions.

Step-by-step explanation:

The table and graph is mentioned below.

In mathematics, a graph is a visual representation of a set of objects, often referred to as vertices or nodes, and the connections or edges that exist between them. Graphs are used to represent a wide range of mathematical concepts, such as functions, equations, and data sets.

A graph can be thought of as a collection of points or vertices that are connected by lines or edges. The way these vertices and edges are arranged can convey important information about the relationships between the objects being represented.

Graphs can be classified into different types, such as directed or undirected graphs, weighted or unweighted graphs, and cyclic or acyclic graphs. They can also be used to model real-world systems, such as social networks, transportation systems, and computer networks.

a)Table for Members:

Number of Movies (n) Cost (C1)

               0                 $75.00

               10                 $95.00

               20                 $115.00

               30                 $135.00

               40                 $155.00

               50                 $175.00

Table for Non-members:

Number of Movies (n) Cost (C2)

                0                 $5.75

                10                 $60.75

                20                 $115.75

                30                 $170.75

                40                 $225.75

                50                 $280.75

b) On the same set of axes, the graph is attached below.

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Different kinds of graphs, including directed and undirected graphs, weighted and unweighted graphs, and cyclic and acyclic graphs, can be distinguished.

In mathematics, a graph is a picture of a collection of things, also known as vertices or nodes, and the links or edges that join them. Numerous mathematical ideas, including functions, formulae, and data sets, are represented by graphs.

A collection of vertices or points linked by lines or edges is referred to as a graph. The arrangement of these vertices and edges can reveal crucial details about the connections between the things being represented.

Different kinds of graphs, including directed and undirected graphs, weighted and unweighted graphs, and cyclic and acyclic graphs, can be distinguished. Additionally, they can be used to simulate real-world systems like computer networks, transit systems, and social networks.

a)Table for Members:

Number of Movies (n) Cost (C1)

0                 $75.00

10                 $95.00

20                 $115.00

30                 $135.00

40                 $155.00

50                 $175.00

Table for Non-members:

Number of Movies (n) Cost (C2)

0                 $5.75

10                 $60.75

20                 $115.75

30                 $170.75

40                 $225.75

50                 $280.75

b) On the same set of axes, the graph is attached below.

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Answer:

Step-by-step explanation:

To find out how much shorter the pencil was after the student completed the project, we need to subtract the final length from the initial length:

7 1/2 - 5 7/8

To subtract these two mixed numbers, we need to convert them to a common denominator. The smallest common multiple of 2 and 8 (the denominators of the fractions) is 8, so we can rewrite the numbers as:

15/2 - 47/8

Now we can find a common denominator of 8:

(15/2) * (4/4) - (47/8) = 30/8 - 47/8

Subtracting the numerators, we get:

-17/8

Therefore, the pencil was 17/8 inches shorter after the student completed the project than when the student started the project. We can simplify this fraction to a mixed number:

-2 1/8

So the answer is C) 2 3/8 inches.

Answer:

  x = 7

Step-by-step explanation:

You want the value of x in the diagram of similar triangles.

Corresponding side segments in the figure are proportional:

  (2x +10)/(3) = (40)/(5)

  2x +10 = 3·8 . . . . . . . . . . multiply by 3

  2x = 14 . . . . . . . . . . . subtract 10

  x = 7 . . . . . . . . . . divide by 2

The value of x is 7.

Answer:

Sure, I can help you with that!

To start, let's look at the ingredient list and see what information we have:

- 8 cups water

- ½ cup garlic

- ½ cup onion

- 1 ½ cups chopped carrots

- ½ cup tomato paste

- 2 cups sliced potatoes

- 1 cup olive oil

To make the calculations, we can set up ratios that relate the amount of each ingredient to the number of servings of soup. For example, if we want to know how much garlic we need for 40 servings of soup, we can set up the following ratio:

½ cup garlic / 10 servings = x cups garlic / 40 servings

To solve for x, we can cross-multiply and simplify:

(½ cup garlic) * 40 servings = (10 servings) * x cups garlic

20 cups = 10x

x = 2 cups

So we would need 2 cups of garlic to make 40 servings of soup.

Similarly, we can set up ratios for the other questions:

Question 2:

½ cup tomato paste / 10 servings = x cups tomato paste / 60 servings

(½ cup tomato paste) * 60 servings = (10 servings) * x cups tomato paste

30 cups = 10x

x = 3 cups

So we would need 3 cups of tomato paste to make 60 servings of soup.

Question 3:

1 ½ cups chopped carrots / 10 servings = x cups chopped carrots / 30 servings

(1 ½ cups chopped carrots) * 30 servings = (10 servings) * x cups chopped carrots

45 cups = 10x

x = 4.5 cups

So we would need 4.5 cups of chopped carrots to make 30 servings of soup.

here I simplified and checked the cross products of the ratios.

Question 1:

½ cup garlic / 10 servings = x cups garlic / 40 servings

Cross-multiplying, we get:

(½ cup garlic) * 40 servings = (10 servings) * x cups garlic

20 cups = 10x

Dividing both sides by 10, we get:

x = 2 cups garlic

Therefore, we need 2 cups of garlic to make 40 servings of soup.

Checking the cross products:

(½ cup garlic) * 40 servings = (10 servings) * 2 cups garlic

20 cups = 20 cups

The cross products are equal, so my answer is correct.

Question 2:

½ cup tomato paste / 10 servings = x cups tomato paste / 60 servings

Cross-multiplying, we get:

(½ cup tomato paste) * 60 servings = (10 servings) * x cups tomato paste

30 cups = 10x

Dividing both sides by 10, we get:

x = 3 cups tomato paste

Therefore, we need 3 cups of tomato paste to make 60 servings of soup.

Checking the cross products:

(½ cup tomato paste) * 60 servings = (10 servings) * 3 cups tomato paste

30 cups = 30 cups

The cross products are equal, so my answer is correct.

Question 3:

1 ½ cups chopped carrots / 10 servings = x cups chopped carrots / 30 servings

Cross-multiplying, we get:

(1 ½ cups chopped carrots) * 30 servings = (10 servings) * x cups chopped carrots

45 cups = 10x

Dividing both sides by 10, we get:

x = 4.5 cups chopped carrots

Therefore, we need 4.5 cups of chopped carrots to make 30 servings of soup.

Checking the cross products:

(1 ½ cups chopped carrots) * 30 servings = (10 servings) * 4.5 cups chopped carrots

45 cups = 45 cups

The cross products are equal, so my answer is correct.