Noah is designing a set for a school theater production. He has 150 cardboard bricks. He needs to use some bricks to make a chimney and 4 times as many bricks to make an arch. He also saves 15 bricks in case some get crushed. How many cardboard bricks can he use to make the arch?

a basis for the eigenspace corresponding to the eigenvalue λ = 1 is { [4, -2] }, and a basis for the eigenspace corresponding to the eigenvalue λ = 2 is { [0, 0] }.

To find a basis for the eigenspace corresponding to each listed eigenvalue of matrix A, we need to solve the system (A - λI)v = 0, where A is the given matrix, λ is the eigenvalue, I is the identity matrix, and v is a non-zero vector in the eigenspace.

Given the matrix A = [[11, 2], [1, 2]], we will find the eigenspaces corresponding to the eigenvalues λ = 1 and λ = 2.

1) For λ = 1:

We need to solve the equation (A - λI)v = 0.

Substituting λ = 1 and I as the 2x2 identity matrix, we have:

(A - I)v = 0

[[11, 2], [1, 2]] - [[1, 0], [0, 1]] = [[10, 2], [1, 1]]v = 0

To find the basis for the eigenspace corresponding to λ = 1, we need to solve the homogeneous system of equations:

10v₁ + 2v₂ = 0

v₁ + v₂ = 0

One solution to this system is v = [2, -1]. However, since we want a basis (more than one vector), we can multiply this solution by any non-zero scalar. Let's multiply by 2:

v₁ = 4, v₂ = -2

Therefore, a basis for the eigenspace corresponding to λ = 1 is { [4, -2] }.

2) For λ = 2:

We need to solve the equation (A - λI)v = 0.

Substituting λ = 2 and I as the 2x2 identity matrix, we have:

(A - I)v = 0

[[11, 2], [1, 2]] - [[2, 0], [0, 2]] = [[9, 2], [1, 0]]v = 0

To find the basis for the eigenspace corresponding to λ = 2, we need to solve the homogeneous system of equations:

9v₁ + 2v₂ = 0

v₁ = 0

From the first equation, we can see that v₁ = 0. Substituting this into the second equation, we have v₂ = 0 as well.

Therefore, a basis for the eigenspace corresponding to λ = 2 is { [0, 0] }.

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In this question, we have to find an equation for two lines, depending on the input x, which creates the following piecewise function for this graph:

\(y = t, 0 \leq t \leq 10, -t + 20, 10 \leq t \leq 20\)

Equation of a line:

The equation of a line is given by:

\(y = mt + b\)

In which m is the slope and b is the y-intercept(value of y when x = 0).

This situation:

x between 0 and 10:

From here, we can take two points (t,y). I will take (0,0) and (10,10).

From point (0,0), we get that when \(x = 0, y = 0\), which means that the y-intercept is \(b = 0\), thus:

\(y_1 = mt\)

To find the slope, when we have two points, it is given by change in y divided by change in t, so:

Change in t: 10 - 0 = 10

Change in y: 10 - 0 = 10

Slope: \(m = \frac{10}{10} = 1\)

Thus, the first definition is:

\(y_1 = t, 0 \leq t < 10\)

x between 10 and 20:

I will take the points (10,10) and (20,0).

First, we find the slope:

Change in t: 20 - 10 = 10

Change in y: 0 - 10 = -10

Slope: \(m = \frac{-10}{10} = -1\)

Thus:

\(y_2 = -t + b\)

For the intercept, we have point (20,0), which means that when \(t = 20, y = 0\). So

\(0 = -20 + b\)

\(b = 20\)

Thus, the second definition is:

\(y_2 = -t + 20, 10 \leq t \leq 20\)

Write an equation for the graph.

We use the two definitions, that is:

\(y = y_1, 0 \leq t \leq 10, y_2, 10 \leq t \leq 20\)

So

\(y = t, 0 \leq t \leq 10, -t + 20, 10 \leq t \leq 20\)

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Answer:

Linear Angles

Step-by-step explanation:

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Answer:

Complementary Angles

Explanation:

Complementary angles combine to form 180 degrees- a straight line.

The first runner covers 30 yards in 4 seconds, a speed of 5 yards per second. The second runner covers a distance of 20 yards in 4 seconds but starts 10 yards ahead. Both runners have a speed of 5 yards per second.

The first runner maintains a constant speed of 5 yards per second and covers a distance of 30 yards in 4 seconds. The second runner also has a speed of 5 yards per second but starts 10 yards ahead. Therefore, the second runner covers a shorter distance of 20 yards in the same 4 seconds.

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Answer:

There are 6 slices on each pizza

Step-by-step explanation:

Because 18 divided by 3 is 6

The equation 3+7+2=+2 is actually not true, but false.

The equation 3+7+2=+2 is actually not true, but false. This is because the sum of 3, 7, and 2 is 12, not 2.

In general, an equation is considered true if the expressions on both sides of the equal sign are equivalent in value. In this case, the expressions on the left-hand side (3+7+2) and the right-hand side (+2) are not equivalent, and therefore the equation is false.

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Answer:

8

Step-by-step explanation:

According to the empirical rule, Probability = approximately 99.7% .

Approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations.

The empirical rule, also known as the 68-95-99.7 rule, provides a way to estimate probabilities based on the standard deviation of a population. Given a population mean (μ) of 7 and a standard deviation (σ) of 2, we can use the empirical rule to find the probabilities for different ranges of values. According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations.

Using the empirical rule, we can estimate the probabilities for different ranges of values based on the given mean (μ) and standard deviation (σ).

Within one standard deviation of the mean:

The range is from μ - σ to μ + σ.

Probability = approximately 68%

Within two standard deviations of the mean:

The range is from μ - 2σ to μ + 2σ.

Probability = approximately 95%

Within three standard deviations of the mean:

The range is from μ - 3σ to μ + 3σ.

Probability = approximately 99.7%

For the given population mean of μ = 7 and a standard deviation of σ = 2, we can use the empirical rule to estimate the probabilities as described above. These probabilities provide a rough estimate of how likely it is for a randomly selected data point to fall within each respective range. Keep in mind that the empirical rule assumes a normal distribution and may not be precise for all data sets.

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There are 720 different ways can a race with 6 runners be completed.

What is a combination?

The process of combining or the condition of combining. a combination of things; an amalgamation of concepts. a grouping of notes: A chord is a grouping of notes. a grouping of people or entities acting together to impede commerce.

Here, we have

The winner can be chosen from all 6 runners, the second person can be chosen from 5 people, the third - from 4, and so on.

So the total number of possible results can be calculated as:

n = 6×5×4×3×2×1= 6!

= 720

Hence, there are 720 different ways can a race with 6 runners be completed.

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Answer:

The positive difference in the 'y' coordinates of where they agreed to meet and where they should actually meet is 4

Step-by-step explanation:

The segement they form is, using the distance formula,

\(d=\sqrt{((x1-x2)^2+(y1-y2)^2)}\)

but since we only need to find the y coordinates,

we do not need it in this case

now ,

in our case, x1 = 5, y1 = -11

x2=-7, y2=13

so,

y1-y2 is the total y difference for the first points,

so d1=-11-13

d1 = -24

but since distance is positive,so,

d1 = 24

and the midpoint of that would be m= d/2,

m1 = 12

if Barbara is at (-5,5), then the distance will be,

since then, x2=-5,y2=5,

d2 = -11-5

d2=-16

or,

d2 = 16

and

m2 = 16/2

m2 = 8

so the positive difference in the 'y' coordinates of where they agreed to meet and where they should actually meet is,

diff = 12-8

diff = 4

so the difference in y coordinates is 4

Answer:

a) \(C(t)=245,000\cdot(1.035)^t\)

b) Nicki's house is worth $281,143 in 4 years

Step-by-step explanation:

Exponential Growth Function

The exponential function is commonly used to model natural growing or decaying processes where the change is proportional to the actual quantity.

An exponential growth function is expressed as:

\(C(t)=C_o\cdot(1+r)^t\)

Where:

C(t) is the actual value of the function at time t

Co is the initial value of C at t=0

r is the growth positive rate, expressed in decimal

Nicki paid Co=$245,000 for a house and it's known that home prices increase by r=3.5%=0.035 each year.

a)

Substituting the given values, the equation is:

\(C(t)=245,000\cdot(1+0.035)^t\)

\(\mathbf{C(t)=245,000\cdot(1.035)^t}\)

b) The value of Nicki's house in t=4 years is:

\(C(4)=245,000\cdot(1.035)^4\)

\(C(4)=245,000\cdot 1.148=281,143\)

Nicki's house is worth $281,143 in 4 years

Answer:

22.5 more calories are burned

Hiking:

y=255

kayak

y=202.5

The point estimators in your list are b. μ (estimated by the sample mean) and c. s (which is an estimator for σ, the population standard deviation).

A point estimator is a statistic that is used to estimate a population parameter. Out of the options provided, the following are point estimators:

b. μ (mu) - This symbol represents the population mean, which is a measure of central tendency for the entire population. A point estimator for μ would typically be the sample mean (x), calculated from a random sample taken from the population.

c. s - This symbol represents the sample standard deviation, which is a measure of how dispersed the data is from the sample mean. The sample standard deviation (s) is a point estimator for the population standard deviation (σ).

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If his beard  was 8 millimeter long and each passing week it grew 2 additional millimeters, the graph that represents the relationship between the length of his beard in millimeter and time in week has been plotted

The initial length of the beard = 8 millimeter

The length of beard grow in each week = 2 millimeter

Consider the number of week as x

Therefore the linear relationship will be

The length of the beard y = 2x + 8

Plot the graph using the equation

Hence, If his beard  was 8 millimeter long and each passing week it grew 2 additional millimeters, the graph that represents the relationship between the length of his beard in millimeter and time in week has been plotted

The complete question is

Rip van Winkle fell asleep for a very long time. When he fell asleep, his beard was 8 millimeters long, and each passing week it grew 2additional millimeters.

Graph the relationship between the length of Rip van Winkle's beard (in millimeters) and time (in weeks).

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Answer:

85:24

Step-by-step explanation:

85 is the dogs and 24 are the other animals in the office

7 fishes οf the new released initially in the given expοnential pοpulatiοn grοwth. (οptiοn c)

Expοnential pοpulatiοn grοwth means that whatever the pοpulatiοn size is, it keeps increasing because the pοpulatiοn per individual remains the same, accelerating pοpulatiοn grοwth that is limited by limited resοurces. Hence, this increase in pοpulatiοn size is knοwn as expοnential pοpulatiοn grοwth.

The grοwth οf their pοpulatiοn is mοdeled by the expοnential functiοn

P(t)=\(7b^{t}\) ,where t is the time in weeks after the release and b is a pοsitive unknοwn base.

Initially that is t=0

sο fοr t=0,

p(0)=7×1 [since sοmething tο the pοwer zerο is equal tο 1]

Hence, P(0)=7

sο there were 7 fishes initially.

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Answer:

14.52% are unicyclists

Step-by-step explanation:

First, we can use proportions to find the number of unicyclists at the convention. Since we know that the ratio of unicyclists to aerial artists is 9:11, and there are 88 aerial artists, we can set up the following equation:

\(\frac{9}{11}=\frac{x}{88}\)

If we cross multiply, we get that 11x = (88)(9). After we divide through by 11 to isolate x, we get that x = (8)(9) = 72

Second, we have to figure out the number of mimes at the convention to figure out the total number of people there. We know that the ratio of unicyclists to mimes is 3:14, and the number of unicyclists is 72. So, we can set up the following proportion:

\(\frac{3}{14}=\frac{72}{y}\)

If we cross multiply, we get that 3y = 1008, or y = 336 mimes

The total number of people at the convention is 336 mimes + 72 unicyclists + 88 unicyclists = 496. Now we have to figure out what percent of 496 is 72 (the number of unicyclists). If we let z = the percentage, we can simply set up an equation that says that 72 is z% of 496:

\(72=0.01z*496\\0.01z=0.14516\\z=14.52\)

This means that approximately 14.52% of the performers are unicyclists.

Answer:

14.52%

Step-by-step explanation:

U/A = 9/11       and   A = 88      multiply by  8/8   ( to get A=88 as given)

9/11 * 8/8 = 72/88  = U / A       so U = 72

Then U/M = 3/14    multiply by  24/24   ( to get U = 72 as found above)

  then  U/M=    72 / 336     M = 336

U / ( A + U + M )  x  100% = 14.52 %

The population of the bacteria culture can be modeled using the differential equation dp/dt = 0.2p, where p(t) is the population at time t. Solving this differential equation, we get p(t) = 20e^(0.2t). Thus, the population of the bacteria after 40 days is approximately 363.9.

(1) To find the population after 40 days, we simply plug in t = 40 into the equation: p(40) = 20e^(0.2*40) = 104857.6. Therefore, the population after 40 days is 104857.6 (do not round).
(2) To find the time it takes for the population to double, we set p(t) = 2*20 = 40 (since the initial population is 20) and solve for t:
40 = 20e^(0.2t)
2 = e^(0.2t)
ln(2) = 0.2t
t = ln(2)/0.2 ≈ 3.5
Therefore, it takes approximately 3.5 days for the population to double.
To find the population of a culture of bacteria after a certain time period, we can use the formula for exponential growth: p(t) = p0 * e^(rt), where p0 is the initial population, r is the growth rate, and t is the time in days.
(1) To find the population after 40 days, we can plug the given values into the formula: p(40) = 20 * e^(0.2*40). Solving for p(40), we get p(40) ≈ 363.933.
(2) To find the time it takes for the population to double, we can set up an equation using the same formula: 2*p0 = p0 * e^(rt). Dividing both sides by p0, we get 2 = e^(rt). We know the growth rate, r, is 0.2, so we can rewrite the equation as 2 = e^(0.2t).
To solve for t, we can take the natural logarithm of both sides: ln(2) = 0.2t. Then, we can isolate t by dividing both sides by 0.2: t = ln(2) / 0.2 ≈ 3.5. Therefore, it takes approximately 3.5 days for the population to double.

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The correct answer is (a) event. An event is a subset of outcomes from the sample space. It represents a specific outcome or set of outcomes that we are interested in. Events can be simple, consisting of a single outcome, or they can be compound, consisting of multiple outcomes.

For example, consider rolling a fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}. Let's say we are interested in the event of rolling an even number. The event in this case would be {2, 4, 6}, which is a subset of the sample space.

Events can also be mutually exclusive, meaning they cannot occur at the same time, or they can be independent, meaning the occurrence of one event does not affect the probability of the other event occurring.

In summary, an event is a subset of outcomes from the sample space and represents a specific outcome or set of outcomes that we are interested in. It is an important concept in probability theory.

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There were 152 inches of ribbon on the full spool when Wanda used 3/4 of a full spool of ribbon for her project.

A proportion is an equation asserting that two ratios are equal. It can be expressed as a/b = c/d, where a and b represent one ratio and c and d represent another ratio. You may use cross-multiplication to solve proportional issues. This entails multiplying the first ratio's numerator by its denominator and setting the result equal to the first ratio's numerator multiplied by its denominator. After that, you may find the unknown variable.

Let us suppose the number of inches = x.

Wanda used 3/4 of the spool, thus the equation of x can be set as:

1/4 x = 38

x = 152

Hence, there were 152 inches of ribbon on the full spool.

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Rachel should perform better because of the spacing effect.

Given,

Rachel will study one hour each day for 5 days and Jennifer will study 5 hours the day before the exam.

What do you predict about their scores?  

Rachel should perform better because of the spacing effect.

The spacing effect demonstrates that learning is more effective when study sessions are spaced out. This effect shows that more information is encoded into long-term memory by spaced study sessions, also known as spaced repetition or spaced presentation, than by massed presentation.

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The real square roots of 1/100 are ±1/10. To find them, take the square root of both sides of the equation x^2 = 1/100, simplify, and consider the positive values.

To find the real square roots of 1/100, we need to find the values of x that satisfy the equation x^2 = 1/100.

1. Take the square root of both sides of the equation:
√(x^2) = √(1/100)

2. Simplify:
|x| = 1/10

3. Since the square root of a number is always positive, we can conclude that the real square roots of 1/100 are ±1/10.

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Answer:

100 dollors

Step-by-step explanation:

The total profit for selling 120 lunches is $310

An equation is a mathematical statement that shows that two mathematical expressions are equal.

Given that, The graph shows a school's profit P for selling x lunches on one day.

Considering two points (50, 0) and (0, -150) we will find the equation of the line,

The equation of a line passing through two points is given by =

y-y₁ = (y₂-y₁) / (x₂-x₁) × (x-x₁)

y-0 = -150 / -50 (x-50)

y = 3x - 150

Here, y is the total profit and x is the number of lunches,

When x = 120

y = 3(120)-50

y = 360-50

y = 310

Hence, the total profit for selling 120 lunches is $310

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Answer:

\(d = 15.620499\)

Step-by-step explanation:

To solve this question, we must use the following distance equation:

\(\sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} }\)

Plug your coordinates into the equation. Remember, a coordinate is denoted as (x, y).

\(\sqrt{(12-2)^{2} + (13-1)^{2}}\)

\(\sqrt{(10)^{2} + (12)^{2}}\)

\(\sqrt{100+144}\)

\(\sqrt{244} = 15.620499\)

Answer:

Rotations, Reflections

Step-by-step explanation:

This is going to be our answer because we know that a figure has some kind of rotation.

A rotation is when a shape can be rotated in some way. A reflection is a transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a plane. When reflecting a figure in a line or in a point, the image is congruent to the preimage. A reflection map every point of a figure to an image across a fixed-line.

So our answer would be Rotations, Reflections!

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After 20 minutes, the temperature of the pan will be approximately 29°F.

Using Newton's Law of Cooling, we can write:

T(t) = To + (T₁ - To)e^(-kt)

where T(t) is the temperature of the pan at time t, To is the temperature of the freezer, T₁ is the initial temperature of the pan, and k is a constant that depends on the properties of the pan and the freezer.

We know that after 15 minutes, the temperature of the pan is 41°F, so we can write: 41 = 0 + (140 - 0)e^(-15k)

Simplifying this expression, we get:

e^(-15k) = 41/140

Taking the natural logarithm of both sides, we get:

-15k = ln(41/140)

Solving for k, we get:

k ≈ 0.056

Now we can use this value of k to find the temperature of the pan after 20 minutes:

T(20) = 0 + (140 - 0)e^(-0.056*20) ≈ 29°F

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After 20 minutes, the temperature of the pan will be approximately 29°F.

Using Newton's Law of Cooling, we can write:

T(t) = To + (T₁ - To)e^(-kt)

where T(t) is the temperature of the pan at time t, To is the temperature of the freezer, T₁ is the initial temperature of the pan, and k is a constant that depends on the properties of the pan and the freezer.

We know that after 15 minutes, the temperature of the pan is 41°F, so we can write: 41 = 0 + (140 - 0)e^(-15k)

Simplifying this expression, we get:

e^(-15k) = 41/140

Taking the natural logarithm of both sides, we get:

-15k = ln(41/140)

Solving for k, we get:

k ≈ 0.056

Now we can use this value of k to find the temperature of the pan after 20 minutes:

T(20) = 0 + (140 - 0)e^(-0.056*20) ≈ 29°F

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It takes the robot approximately 5.71/1 = 5.71 hours to complete one task. The robot can complete approximately 0.175 tasks in one hour.

To find how long it takes the robot to complete one task, we can divide the total time by the number of tasks. Since the robot can complete 4 tasks in 7/10 hour, we can set up the following equation:

4 tasks / (7/10 hour) = x tasks / 1 hour

Simplifying this equation, we get:

(4 / 0.7) = (x / 1)

Multiplying both sides by 1, we get:

x = 5.71

To find how many tasks the robot can complete in one hour, we can divide the total time available (1 hour) by the time it takes to complete one task. From the calculation above, we know that it takes 5.71 hours to complete one task, so we can set up the following equation:

1 hour / (5.71 hours / task) = x tasks / 1 hour

Simplifying this equation, we get:

(1 / 5.71) = x

Multiplying both sides by 1 hour, we get:

x = 0.175 tasks/hour

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