At which values in the interval [0, 2π) will the functions f (x) = 2cos2θ and g(x) = −1 − 4cos θ − 2cos2θ intersect?
a: theta equals pi over 3 comma 4 times pi over 3
b: theta equals pi over 3 comma 5 times pi over 3
c: theta equals 2 times pi over 3 comma 4 times pi over 3
d: theta equals 2 times pi over 3 comma 5 times pi over 3

The equations of straight lines are \hat{y} = 2h, \hat{y} = 23r + 4 and \hat{y}= 2s + 3t. Option(A),(B) and (F) are correct.

A line in the coordinate plane can be described with the help of a linear equation, that is, an equation that has a first-degree expression, like y = 2x – 3.

There are many ways to put the equation of a line in the form y = mx + b,

where m is the slope and

b is the y-intercept,

but they all require the use of algebraic properties of equations, such as addition, subtraction, multiplication, division, and substitution.

The equations of straight lines among the following are:  \hat{y} = 2h,  \hat{y} = 23r + 4 and \hat{y}= 2s + 3t

Hence, the correct options are:Option A: \hat{y} = 2h , Option B: \hat{y} = 23r + 4  and Option F: \hat{y}= 2s + 3t.

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Answer:

73°

Step-by-step explanation:

As it is a hexagon, the sum of the interior angles in 720°

4x+165+132+131=720 4x=292 x=73°

The length of the side opposite the 85° angle is approximately 7.75 units.

To solve this problem, we can use the law of sines, which states that for any triangle ABC:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the side lengths of the triangle opposite the angles A, B, and C, respectively.

In this case, we know the side length b = 5 and the angles A = 85° and B = 40°.

Let's call the unknown side length opposite the angle A as a.

Then, we have:

a/sin(85°) = 5/sin(40°)

Multiplying both sides by sin(85°), we get:

a = 5*sin(85°)/sin(40°)

Using a calculator, we get:

a ≈ 7.75

Therefore, the side length opposite the angle of 85 degrees is

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Complete question

A triangle has side of 5, angle of 85*, and an angle of 40 opposite the side length of 5. What is the length of the side opposite the 85° angle?

Answer:

The sum of the first n terms in an arithmetic sequence is (n/2)⋅(a₁+aₙ). It is called the arithmetic series formula.

Step-by-step explanation:

An arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms. Following is a simple formula for finding the sum: Formula 1: If S nrepresents the sum of an arithmetic sequence with terms , then. This formula requires the values of the first and last terms and the number of terms.

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The perimeter of the figure is 16p + 10.

The whole length of a two-dimensional or three-dimensional shape's sides or edges is known as its perimeter. It is frequently referred to as the shape's perimeter or circumference. It is possible to determine the perimeter of many geometric forms with accuracy. The perimeter is a key idea in geometry and has several practical uses, such as determining the radius of a circular racetrack or determining the length of fencing required for a certain property.

The perimeter of a figure is the sum of the lengths of all its sides.

Thus,

Perimeter = (p - 9) + (7p + 5) + (p - 9) + (7p + 5)

Perimeter = 2p + 2(7p) + 2(5)

Perimeter = 16p + 10

Hence, the perimeter of the figure is 16p + 10.

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The number we are looking for is N = 2 × 2^3 × 3^2 = 72.

Let's first recall some properties of the number of divisors of an integer. If we factorize an integer n as a product of prime powers, say

n = p_1^a_1 × p_2^a_2 × ... × p_k^a_k

then the number of divisors of n is given by

d(n) = (a_1 + 1) × (a_2 + 1) × ... × (a_k + 1).

Using this fact, we can deduce some information about the number we are looking for. Let's call it N. We know that N has 8 divisors, so it must be of the form

N = p_1^2 × p_2^2, or N = p_1^7,

where p_1 and p_2 are distinct prime numbers.

Now, we are told that each of N/2 and N/3 has four divisors. We can use the same fact about the number of divisors to conclude that

N/2 = q_1^3 × q_2, or N/2 = q_1^1 × q_2^3,

and

N/3 = r_1^3 × r_2, or N/3 = r_1^1 × r_2^3,

where q_1, q_2, r_1, and r_2 are distinct prime numbers.

To simplify the notation, let's introduce the variables a, b, c, d, e, and f, defined by

p_1 = q_1^a × q_2^b,

p_2 = r_1^c × r_2^d,

N/2 = q_1^e × q_2^f,

N/3 = r_1^g × r_2^h.

Using the information we have so far, we can write down equations for a, b, c, d, e, f, g, and h in terms of unknown exponents:

a + 1 × (b + 1) = e + 1 × (f + 1) = 4,

c + 1 × (d + 1) = g + 1 × (h + 1) = 4,

2a × 2b = ef,

2c × 2d = gh.

We can solve this system of equations by trial and error. For example, we can start by trying all possible values of a and b such that 2a × 2b = 4. This gives us two possibilities: a = 0, b = 2, or a = 1, b = 1. Using the first possibility, we get e = 3, f = 1, which leads to N/2 = q_1^3 × q_2, and hence N = 2 × q_1^3 × q_2^2. Substituting this into the equation for the sum of divisors, we get

(1 + q_1 + q_1^2 + q_1^3) × (1 + q_2 + q_2^2) = 216.

We can solve this equation by trial and error as well, or by observing that 216 = 2^3 × 3^3, and hence the two factors on the left-hand side must be equal to 2^3 and 3^3, respectively. This gives us the unique solution q_1 = 2 and q_2 = 3, and hence N = 2 × 2^3 × 3^2 = 72.

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After solving the given problem the value of tan N rounded to the nearest hundredth is 0.95.

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of the triangles.

It is used to calculate distances, heights, and angles, and is used extensively in fields such as engineering, physics, and architecture.

We can use the tangent function to find the value of tan N in the right triangle LMN.

tan N = ML/MN

We know that MN = 6 and ∠LMN is 90°, so we can use the Pythagorean theorem to find ML:

NL² = ML² + MN²

√77² = ML² + 6²

77 = ML² + 36

ML²= 77 - 36

ML² = 41

ML = √41

Now we can substitute the values for ML and MN into the equation for tangent:

tan N = ML/MN = (√41)/6

Using a calculator, we can evaluate this expression and round to the nearest hundredth:

tan N ≈ 0.95

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The complete question is given below.

(a) False; A subspace is formed by the set of vectors that do not belong to the column space.

(b) True; If the matrix contains solely the zero vector, then it is the zero matrix.

(c) True; The column space of a particular matrix is equivalent to the column space of another specified matrix.

(d) False; The column space of one matrix is identical to the column space of another matrix.

(a) False; if A = [1 0; 0 0], then the column space of A is { e1 }, where e1 is the standard unit vector in the plane. If v is not in the column space of A, but w is not in the column space of A, then v + w is not in the column space of A.

Therefore, the set of vectors that are not in the column space of A does not form a subspace.

(b) True; if every vector in Rn is in the null space of A, then in particular, every standard unit vector is in the null space of A. Thus, the ith column of A is zero for i = 1, . . . , n, so A is the zero matrix.

(c) True; the column space of A is generated by the columns of A, while the column space of AB is generated by linear combinations of the columns of AB. By definition of matrix multiplication, the columns of AB are linear combinations of the columns of A, so the column space of AB is a subspace of the column space of A. Conversely, let b be in the column space of A. Then there is an x in Rm such that Ax = b. Thus, ABx = A(Bx), so b is in the column space of AB. Therefore, the column space of A is a subspace of the column space of AB. Hence the two column spaces are equal.

(d) False; if A = [1 0; 0 0] and B = [0 0; 0 1], then the column space of A is { e1 }, while the column space of B is { e2 }. The column space of AB is { 0 }, so it is not equal to either column space.

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Using Law of Cosines, the square root of a negative number is not a real number, there is no solution for x.

We can use the Law of Cosines to solve this problem. Let's label the distance between Samir and Ayisha as "x". Then, using the Law of Cosines, we have:

cos(18°) = (80² + x² - 82²) / (2 * 80 * x)

Simplifying this equation, we get:

cos(18°) = (x² - 2x + 3996) / (160x)

Multiplying both sides by 160x, we have:

160x * cos(18°) = x² - 2x + 3996

Simplifying, we get:

0 = x² - 2x + 3996 - 160x * cos(18°)

Now we can use the quadratic formula to solve for x:

[tex]x = [2 \pm \sqrt{(4 - 4(1)(3996 - 160cos(18^o))) ]} / 2[/tex]

[tex]x = [1 \pm \sqrt{(1 - (3996 - 160cos(1^o°)))} ]\\x = [1 \pm \sqrt{(160cos(18^o) - 3995)]}[/tex]

Plugging in the value of cos(18°) (which is approximately 0.951), we get:

[tex]x = [1 \pm \sqrt{(160(0.951) - 3995)}]x = [1 \pm \sqrt{(-2494.4)}][/tex]

Since the square root of a negative number is not a real number, there is no solution for x. Therefore, the problem may be incorrect or incomplete.

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Assuming a 0.05 significance level, both hypothesis tests would reject the null hypothesis. This means that there is enough evidence to suggest that the shopping cart icon's design affects the users' behavior. Therefore, the design should be changed to improve the user's experience.

Step by step explanation:

The problem is not stated, so we need to make some assumptions. We will assume that the hypothesis tests were done to determine whether a change in the shopping cart icon design would affect the users' behavior.

The null hypothesis (H0) is that the design of the shopping cart icon does not affect the users' behavior, while the alternative hypothesis (Ha) is that the design of the shopping cart icon affects the users' behavior.

The significance level is 0.05, which means that we are willing to accept a 5% chance of making a type I error (rejecting the null hypothesis when it is actually true).

If both hypothesis tests reject the null hypothesis, it means that the p-values were less than 0.05, and there is enough evidence to suggest that the design of the shopping cart icon affects the users' behavior.

Therefore, it is recommended to proceed with changing the design of the shopping cart icon to improve the user's experience.

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The angle between 0 and 360° swept counterclockwise from the 3 o'clock position on the unit circle is  33.69°.

For point D with a y-coordinate 0.5 in Quadrant I, we can use the trigonometric functions to find the angle it makes with the positive x-axis. Since the point is on the unit circle, we have:

x^2 + y^2 = 1

Substituting y = 0.5, we get:

x^2 + (0.5)^2 = 1

x^2 = 1 - (0.5)^2

x = ±√(1 - 0.25)

x = ±√0.75

Since the point is in Quadrant I, we know that x is positive. Therefore, x = √0.75. Now, we can use the inverse tangent function to find the angle θ that the point makes with the positive x-axis:

θ = tan^(-1)(y/x)

θ = tan^(-1)(0.5/√0.75)

θ ≈ 33.69°

Since the point is in Quadrant I, the angle swept counterclockwise from the 3 o'clock position is simply θ:

Angle = 33.69°

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When the effects of x1 and x2 on the dependant variable are influenced by each other in a first-order model with two predictors, x1 and x2, an interaction term may be utilised. The right response is d) both b and c.

A first-order model is a linear equation that involves only one dependent variable and one independent variable. In other words, the equation represents the linear relationship between two variables. The equation can be defined as

Y = β0 + β1X,

where

Y is the dependent variable,

X is the independent variable,

β0 is the y-intercept, and

β1 is the slope of the line.

When attempting to predict the dependant variable based on the independent variable and a linear relationship between them, the first-order model is helpful.

An interaction term is added to a first-order model with two predictors when the effect of x1 on the dependent variable is influenced by x2 as well as the effect of x2 on the dependent variable is influenced by x1. Therefore, the correct answer choice is d) both b and c.

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Answer: The correct answer to this problem would be C, or 32/3.

Step-by-step explanation:

The average rate of change in a function is the rate at which one value changes with respect to another value.

Given function: f(x) = 2x² + x + 5 on interval [-2, 2]

Therefore,

Average rate of change = f(2)-f(-2)/2-(-2)

f(2) = 2(2)² + 2 + 5

f(2) = 15

f(-2) = 2(-2)² - 2 + 5

f(2) = 11

Substitute the values,

Average rate = 5-11/2+2

Average rate = 4/4

Average rate = 1

So, the average rate of change is

Leading me to believe option C is correct.

Answer:

Step-by-step explanation:

Let's represent the amount of money Tara received by "7x", since the ratio of Tara's share is 7:8. Similarly, let's represent the amount Jack received by "8x".

We know that Jack received £18 more than Tara, so we can set up an equation:

8x - 18 = 7x

Solving for x, we get:

x = 18

Now we can find Tara's share by substituting x into our expression for Tara's share:

7x = 7(18) = £126

Therefore, Tara received £126.

In response to the stated question, we may state that The coefficient equation  of -1.327 indicates that the amount of water in the cistern declines at an exponential rate as time passes.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the number "9". The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. In the equation "x2 + 2x - 3 = 0," for example, the variable x is raised to the second power. Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

The least-squares regression line has the following equation:

2.993 - 1.327 In(Volume)* (Time)

The link between the natural logarithm of water volume and the natural logarithm of time is illustrated by this equation. The coefficient of -1.327 indicates that the amount of water in the cistern declines at an exponential rate as time passes.

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The probability of selecting the correct four numbers out of 59 is solved by the formula :

P(4 correct numbers) = (number of ways to choose 4 correct numbers) / (total number of possible 4-number combinations)

The total number of possible 4-number combinations out of 59 is:

C(4,59) = (59 choose 4) = 190,578

P(jackpot) = P(4 correct numbers) * P(1 correct number)

P(jackpot) = 1/41

thus, the probability of winning the jackpot in this particular lottery is 1/41.'

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Answer:0.001204

Step-by-step explanation:

0.028x0.043

The chance that a hypothesis test will incorrectly reject the null hypothesis when the p-value cutoff is 1%. The correct option is A.

A hypothesis test is used to test the validity of a hypothesis by calculating the probability that a sample statistic occurred by chance.

The null hypothesis is the hypothesis that is being tested, and it is usually assumed to be true unless there is evidence to the contrary.

The p-value is the probability of obtaining a sample statistic at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level, which is the p-value cutoff chosen by the researcher, then the null hypothesis is rejected.

If the p-value is greater than or equal to the significance level, then the null hypothesis is not rejected. The p-value cutoff is the significance level chosen by the researcher, and it represents the maximum probability of rejecting the null hypothesis when it is actually true.

In this case, the p-value cutoff is 1%, which means that the maximum probability of rejecting the null hypothesis when it is actually true is 1%. Therefore, the chance that a hypothesis test will incorrectly reject the null hypothesis when the p-value cutoff is 1%.

Therefore, the correct option is A.

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Step-by-step explanation:

This is an equation for a LINE.....ANY point on the line will satisfy this equation....so there are INFINITE solutions.

The rounded value of 25.71 to 2 decimal places is 25.71 itself.

To round 25.71 to 2 decimal places, we need to look at the third decimal place, which is 1.

If the third decimal place is 5 or greater, we round up the second decimal place. If the third decimal place is less than 5, we simply drop it and keep the second decimal place as is.

In this case, the third decimal place is 1, which is less than 5, so we simply drop it and keep the second decimal place as is. Therefore, 25.71 rounded to 2 decimal places is:

25.71 ≈ 25.71 (no rounding necessary)

So, the rounded value of 25.71 to 2 decimal places is 25.71 itself.

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By applying the triangle inequality theorem, the possible whole number measures of the third side in ascending order, are 5, 6, and 7.

The Triangle Inequality Theorem is a theorem that states that the sum of the lengths of two sides of a triangle is greater than the length of the third side.

We can use the triangle inequality theorem to determine the possible whole number measures of the third side of a triangle if the first two sides measure 6 and 2.

The Triangle Inequality Theorem can be written as:

a + b > c where a, b, and c are the sides of a triangle. If a = 6 and b = 2, then the inequality is:

6 + 2 > c

8 > c

So, the third side must be less than 8 units long. On the other hand, the length of the third side must be greater than the difference between the lengths of the other two sides. So, we can set up another inequality:

6 - 2 < x

Simplifying, we get:

4 < x

So, the length of the third side must be greater than 4 units long.

Therefore, applying the triangle inequality theorem, the possible whole number measures of the third side of the triangle are the integers from 5 to 7, inclusive.

So, the list of possible whole number measures of the third side in ascending order is 5, 6, and 7.

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If the null hypothesis were correct, there would be a 0.47 percent chance of getting a sample proportion that deviates from the population proportion's hypothesised value

Proportion refers to the relationship between two quantities or numbers, indicating how they are related to each other in size or amount.

The best interpretation of a Z score of 0.47 for a sample proportion is:

b. If the null hypothesis were correct, there would be a 0.47 percent chance of getting a sample proportion that deviates from the population proportion's hypothesised value.

This interpretation is in line with the definition of a Z score, which is a measure of how many standard deviations a sample statistic (in this case, the sample proportion) is not what would be anticipated if the null hypothesis were true. A Z score of 0.47 means that the sample proportion is 0.47 standard deviations away from the expected value under the null hypothesis. Therefore, the interpretation that the probability of obtaining a sample proportion as far or farther than observed from the hypothesized value of the population proportion is 0.47, assuming the null hypothesis is true, is the most appropriate.

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The simple exponential smoothing model can be best described as (B) "an expression combining the most recent forecast and actual data value". B is the correct answer.

Simple exponential smoothing is a time series forecasting technique that creates predictions using a weighted average of previous observations. As the observations get older, the weights decrease exponentially. The forecast for the next period is created by fusing the most recent forecast with the most recent actual data value using the smoothing parameter alpha. This can be mathematically stated as:

F_t+1 = αY_t + (1-α)F_t

where F_t is the forecast for period t, Y_t is the actual value for period t, α is the smoothing parameter, and F_t+1 is the forecast for period t+1.

Option B is the correct answer.

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In response to the stated question, we may state that As a result, the rectangular pyramid has a surface area of 296 ft2. 296 ft2 is the correct answer.

In Euclidean geometry, a rectangle is a parallelogram with four small angles. It may also be defined as a fundamental rule hexagon or one in which all of the angles are equal. Another alternative for the parallelogram is a straight angle. Four of the vertices of a square are the same length. A quadrilateral with four 90° angle vertices and equal parallel sides has a rectangle-shaped cross section. As a result, it is also known as a "equirectangular rectangle" at times. A rectangle is sometimes referred to as a parallelogram due to the equal and parallel dimensions of its two sides.

To determine the surface area of a rectangular pyramid, first calculate the area of each face and then add them together.

Given the rectangular pyramid's dimensions:

Slant height= 10 ft base length (l) = 12 ft base width (w) = 8 foot

Then, calculate the area of the base. The area of the base is simply l w: because it is a rectangle.

Base area = [tex]12 ft x 8 ft = 96 ft^2[/tex]

A triangle face's area equals 1/2 its base height, where base = w or l and height = sh.

[tex]1/2 *8 ft *10 ft = 40 ft^2[/tex] area of the first triangle face

[tex]1/2 *12 ft *10 ft = 60 ft^2[/tex]area of the second triangular face

Third triangular face area = [tex]1/2 *8 ft *10 ft = 40 ft^2[/tex]

The area of the fourth triangular face is equal to [tex]1/2 *12 ft *10 ft = 60 ft^2.[/tex]

Now put the areas of all the faces together to get the entire surface area:

Total surface area = base area + sum of all triangular face areas

Surface area total =[tex]96 ft^2 + (40 ft^2 + 60 ft^2 + 40 ft^2 + 60 ft^2)[/tex]

296 ft2 total surface area

As a result, the rectangular pyramid has a surface area of 296 ft2.

296 ft2 is the correct answer.

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The statement "Box 1 is negatively skewed, and Box 2 is symmetric" is true.

Looking at shown the box-and-whisker plots, we can make the following observations:

Box 1 has a longer whisker on the left side than on the right side, which indicates that the data is skewed to the left.

Therefore, Box 1 is negatively skewed.

Box 2 has whiskers that are approximately the same length on both sides, which indicates that the data is symmetric.

Therefore, Box 2 is symmetric.

Based on these observations, we can conclude that:

A) Box 1 is negatively skewed, and Box 2 is symmetric.

Therefore, the correct answer is A.

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The missing figure is attached below.

Using distance between two points formula, a = 0

The distance between two points (x, y) and (x', y') is given by

d = √[(x' - x)² + (y' - y)²]

Now, If the distance between the points (0, 6) and (a, 0) is 6 units, to find the value of a,

Let

So, substituting the values of the variables into the equation, we have

d = √[(x' - x)² + (y' - y)²]

6 = √[(a - 0)² + (0 - 6)²]

⇒ √[a² + (- 6)²] = 6

√[a² + 36] = 6

Squaring both sides, we have that

a² + 36  = 6²

a² + 36  = 36

a² = 36 - 36

a² = 0

a = √0

a = 0

So, a = 0

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The question is incomplete. Here is the complete question

If the distance between (0, 6) and (a, 0) os 6 units. Find the value of a

​

based on these assumptions, the country will first experience shortages of food in approximately year 25 (measured from the present).

We can use the concept of exponential growth to model the population and food supply of the country.

Let P(t) be the population of the country at time t (measured in years since the present), and F(t) be the amount of food supply (measured in units of people) at time t. Then, we have:

P(0) = 4 million (initial population)

P(t) = P(0) × (1 + r) raise to the power of t, where r is the annual growth rate of the population (5%) and t is the time in years since the present

F(0) = 8 million (initial food supply)

F(t) = F(0) + p × g, where p is the population growth rate (5%) and g is the additional food supply per person (0.25)

We want to find the year when the food supply first becomes inadequate for the population. This happens when:

P(t) > F(t)

Substituting the expressions for P(t) and F(t), we get:

P(0) × (1 + r)t > F(0) + p × g × t

Simplifying this inequality, we get:

4 × (1.05)t > 8 + 0.25t

Dividing both sides by 4, we get:

(1.05)t > 2 + 0.0625t

We can use trial and error or a graphing calculator to solve this inequality. One way is to try different values of t until we find a value that satisfies the inequality. For example, we can try t = 20:

(1.05)²⁰ = 2.6533...

2 + 0.0625 × 20 = 3.25

Since (1.05)²⁰ < 2 + 0.0625 × 20, this means that the food supply is still adequate at t = 20.

We can try larger values of t until we find a value that satisfies the inequality. For example, we can try t = 25:

(1.05)²⁵ = 3.386...

2 + 0.0625 × 25 = 3.15625

Since (1.05)²⁵ > 2 + 0.0625 × 25, this means that the food supply is no longer adequate at t = 25.

Therefore, According to these projections, the nation's first food shortages will occur around the year 25. (measured from the present).

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Answer:

C. Rachel is saving $5 per week.

Step-by-step explanation:

The initial savings are $10, as it is the y-intercept.

And to obtain the slope we can take 2 points from the graph.

A(0,10)

B(1,15)

m=(y2-y1)/ (x2-x1)

m=(15-10)/ (1-0)

m= 5/1

m= 5 savings in dollars per (1) week

the price of a burrito is $3.50.

How to solve?

Let the price of a burrito be denoted by "b" and the price of a taco be denoted by "t". Then we can set up the following system of equations:

2b + 5t = 19.50

5b + 2t = 22.50

We can solve for one variable in terms of the other in one of the equations, and substitute that expression into the other equation. For example, we can solve the first equation for b:

2b + 5t = 19.50

2b = 19.50 - 5t

b = (19.50 - 5t)/2

Now we can substitute this expression for b into the second equation:

5b + 2t = 22.50

5[(19.50 - 5t)/2] + 2t = 22.50

Simplifying and solving for t:

(97.50 - 25t)/2 + 2t = 22.50

97.50 - 25t + 4t = 45

-21t = -52.50

t = 2.50

So the price of a taco is $2.50. We can substitute this value back into one of the original equations to solve for the price of a burrito. Using the first equation:

2b + 5t = 19.50

2b + 5(2.50) = 19.50

2b + 12.50 = 19.50

2b = 7

b = 3.50

So, the price of a burrito is $3.50.

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