a sample of a radioactive substance decayed to 95% of its original amount after a year. (round your answers to two decimal places.) (a) what is the half-life of the substance?

f(x) is continuous for all values of x, as it can be evaluated for any real number, including both rational and irrational numbers.

f(x) is continuous for all values of x. This means that the function can be evaluated for any real number, including both rational and irrational numbers.

For rational numbers, f(x) = 0. This is because rational numbers can be expressed as a fraction, and when a fraction is divided by itself, the result is always 1. We can express this mathematically as f(x) = x/x = 1 * 0 = 0.

For irrational numbers, f(x) = 7. This is because an irrational number cannot be expressed as a fraction, and so when an irrational number is divided by itself, the result is undefined. In this case, we can define f(x) as 7, since it is a constant value that is independent of the value of x. We can express this mathematically as f(x) = x/x = undefined * 7 = 7.

Therefore, f(x) is continuous for all values of x, as it can be evaluated for any real number, including both rational and irrational numbers.

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(a) With 10 rectangles using left endpoints, 5 rectangles are contributing negative area values to the estimated net area. This means that the function is below the x-axis in those intervals.

The remaining 5 rectangles are contributing positive area values, as the function is above the x-axis in those intervals.

When using midpoints, the number of positive and negative rectangles may not be the same. It depends on the behavior of the function within each subinterval. The use of midpoints can result in a different distribution of positive and negative rectangles compared to using left endpoints.

(b) The error when using midpoints with 10 subintervals can be determined by calculating the difference between the estimated net area using midpoints and the actual net area.

To calculate the error, we need to evaluate the definite integral of the function over the given interval and subtract the estimated net area using midpoints.

Error = Actual Net Area - Estimated Net Area using Midpoints

Since the exact values are not provided, the specific error value cannot be determined without further information or calculations.

Using left endpoints with 10 rectangles, 5 rectangles contribute negative area values and 5 contribute positive area values. When using midpoints, the distribution of positive and negative rectangles may differ. The error when using midpoints can be calculated by subtracting the estimated net area using midpoints from the actual net area, but the exact error value cannot be determined without further information or calculations.

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Answer: 3 boxes

Step-by-step explanation: Divide 21 and 7.

The length of the curve x = 1/3 √y (y − 3)  is 64/3.

The distance between two point along a segment of a curve is known as the arc length.

The equation of the curve is given by:

x = 1/3 √y (y − 3), 16 ≤ y ≤ 25

Length of the curve y = f(x) between point  x =a to x = b is given by:  

\(\int\limits^b_a {\sqrt{1+[f'(x)]^2} } \, dx\)

 √1 + [f′(x)]2 dx.

x = 1/3 √y (y − 3)

x = 1/3 * \(y^\frac{3}{2}\) - \(y^\frac{1}{2}\)

Let's find the first derivative of x.

dx/dy = 1/3. 3/2.  \(y^\frac{1}{2}\)- 1/2 .  \(y^\frac{1}{2}\)

dx/ dy = 1/2 ( \(y^\frac{1}{2}\) - \(y^\frac{-1}{2}\))

(dx/dy)^2= 1/4 (  \(y^\frac{1}{2}\) - \(y^\frac{-1}{2}\))^2

= 1/4(y + \(y^-^1\)-2)

1 + {f′(x)}2 = 1 + 1/4(y +\(y^-^1\)-2)

= 1/4 y + 1/4\(y^-^1\)+ 1/2

1 + {f′(x)}2= 1/4 (y + \(y^-^1\) + 2)

√[1 + {f′(x)}2] = 1/2 (  \(y^\frac{1}{2}\) +  \(y^\frac{-1}{2}\))

Length of curve  is given by:  

25

∫   \(\frac{\sqrt{y} }{2} +\frac{1}{2\sqrt{y} }\) dy

16

= \(\left \{ {{y=25} \atop {x=16}} \right.\) [\(y^\frac{3}{2}\)/3 + √y]

= [(25)3/2 /3 + √25] - [(16)3/2 /3 + √16]

= [125/3 + 5] - [64/3 + 4]

= 140/3 - 76/3

= (140 - 76)/3

= 64/3

So the length of the curve is 64/3

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The mean of the given set of data will be 8.3.

Given is the following data -

3, 5, 8, 11, 11, 12

We have -

3, 5, 8, 11, 11, 12

We can calculate the mean as -

m = (3 + 5 + 11 + 8 + 11 + 12)/6

m = 50/6

m = 8.3

Therefore, the mean of the given set of data will be 8.3.

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Answer:

  14.5 square units

Step-by-step explanation:

You want the area of the triangle inscribed in the 4×9 rectangle shown.

Pick's theorem tells you the area can be found using the formula ...

  A = i +b/2 -1

where i is the number of interior grid points, and b is the number of grid points on the boundary. This theorem applies when the vertices of a polygon are at grid intersections.

The first attachment shows there are 14 interior points, and 3 boundary points. Then the area is ...

  A = 14 + 3/2 -1 = 14 1/2 . . . . square units

The area of the triangle is 14.5 square units.

The area of a triangle can also be found from the determinant of a matrix of its vertex coordinates. The second attachment shows the area computed for vertex coordinates A(0, 4), C(7, 0) and B(9, 3).

The area of the triangle is 14.5 square units.

__

Additional comment

The area can also be found by subtracting the areas of the three lightly-shaded triangles from that of the enclosing rectangle. The same result is obtained for the area of the inscribed triangle.

The area value shown in the first attachment is provided by the geometry app used to draw the triangle.

We find the least work is involved in counting grid points, which can be done using the given drawing.

<95141404393>

Answer:

In order to use the Pythogrean theorem you must first use the points to form a triangle then count the side length in order to get your numbers. It should be 7^2+2^2=c^2 and your answer should be 7.28. This person subtracted the x coordinates from the x coordinate and the y from the y to get their numbers.

Step-by-step explanation:

The probability of A1 occurring given B1 is approximately 0.375, rounded to three decimal places.

To find the probability of P(A1|B1), we can use Bayes' theorem:

P(A1|B1) = (P(B1|A1) * P(A1)) / P(B1)

Given that A1 and A2 are complements, P(A2) can be calculated as 1 - P(A1), which means P(A2) = 0.7.

We are given P(B1|A1) = 0.6 and P(B1|A2) = 0.5.

Now, to calculate P(B1), we can use the law of total probability:

P(B1) = P(B1|A1) * P(A1) + P(B1|A2) * P(A2)

Substituting the given values, we get:

P(B1) = (0.6 * 0.3) + (0.5 * 0.7)

= 0.18 + 0.35

= 0.53

Finally, we can calculate P(A1|B1) using Bayes' theorem:

P(A1|B1) = (P(B1|A1) * P(A1)) / P(B1)

= (0.6 * 0.3) / 0.53

≈ 0.375

Therefore, the probability of A1 occurring given B1 is approximately 0.375, rounded to three decimal places.

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This approach ensures that each house is in range of at least one tower while minimizing the number of towers needed to cover all houses.

To determine the minimum number of towers needed to cover all houses, we can start by sorting the house locations in ascending order. Then, we can place the first tower at x1+r, which covers all houses to the left of x1+r. We can continue placing towers at the first house that is not covered by the previous tower, with its location being the maximum distance from the previous tower that can still cover all houses within its radius.

Specifically, for any house x[i], we can check if it is within the radius of the current tower. If it is not, we place a new tower at x[i-1]+r and repeat the process. Once all houses are covered, we can return the number of towers used and their locations. This algorithm has a time complexity of O(n log n) due to the initial sorting step but can be optimized to O(n) by using a linear search instead.

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The flagpole is 6 yards tall

The height of Carla from the ground, H = 15 yards

Distance between the curb and the buiding, D = 20 yards

The larger triangle is shown below:

A diagram representing the smaller triangle is shown below:

Since She lines up the top of a flagpole with the curb, the two triangles are similar and the ratio of corresponding sides will be equal

Therefore:

15/x = 20/8

20x = 15(8)

20x = 120

x = 120/20

x = 6

The flagpole is 6 yards tall

Answer:

Step-by-step explanation:

how the heck are we supposed to know this?????

Answer:

You use multiplyers

Answer:

Step-by-step explanation:

Danny's speed is given:

Andrew's speed is found from the table.

Since its the direct relationship, we find it using one pair of values:

Danny is faster as 15 mph > 14.5 mph

Speed =Distance/Time

Here speed is equal to slope of box

Danny is faster as his one is 15mph

The velocity, acceleration, and speed of the particle are
v(t) = 2ti + 2j + (7/t)k, a(t) = 2i - (7/t^2)k, s(t) = √((4t^4 + 4t^2 + 49) / t^2) respectivelty.

To find the velocity, acceleration, and speed of a particle with the given position function, we can differentiate the position function with respect to time. Let's go step by step:

1. Position function:
The given position function is r(t) = t^2i + 2tj + 7ln(t)k.

2. Velocity:
To find the velocity, we need to differentiate the position function with respect to time.
Velocity (v) is the derivative of position (r) with respect to time (t).

Taking the derivative of each component of the position function:
d/dt (t^2) = 2t
d/dt (2t) = 2
d/dt (7ln(t)) = 7/t

So, the velocity function (v) is:
v(t) = 2ti + 2j + (7/t)k

3. Acceleration:
To find the acceleration, we need to differentiate the velocity function with respect to time.
Acceleration (a) is the derivative of velocity (v) with respect to time (t).

Taking the derivative of each component of the velocity function:
d/dt (2t) = 2
d/dt (2) = 0
d/dt (7/t) = -7/t^2

So, the acceleration function (a) is:
a(t) = 2i + 0j - (7/t^2)k
or simply,
a(t) = 2i - (7/t^2)k

4. Speed:
The speed of a particle is the magnitude of its velocity vector.
The magnitude of a vector is found using the Pythagorean theorem.
For the velocity function v(t) = 2ti + 2j + (7/t)k, the magnitude of v(t) is:

|v(t)| = √((2t)^2 + (2)^2 + (7/t)^2)
      = √(4t^2 + 4 + 49/t^2)
      = √((4t^4 + 4t^2 + 49) / t^2)

Therefore, the speed function (s) is:
s(t) = √((4t^4 + 4t^2 + 49) / t^2)

This is the velocity, acceleration, and speed of the particle with the given position function.

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Answer:

every other term is negative, so it would be -64

Step-by-step explanation:

hope it helps!

y(x)  = \(\frac{1}{3-x^2}\) is the solution of the given differential equation.

A differential equation is an equation that relates a function and its derivatives; it describes how a quantity changes with respect to another. and its solution is a function that satisfies the equation for all values of the independent variable

y'=2x\(y^2\)

where y' is dy/dx

so dy/dx = 2x\(y^2\)

=> dy/ \(y^2\) =2xdx

integrating on both side we get:

-1/y = \(x^2\) +c   --- (i)

where c is a constant

given that y(1) =2

putting the value of y = 1/2 and x=1 in equation (i) we get

so  -2 = 1+c

=>  c= -3

y(x) = 1/(3-\(x^2\))  is the required solution of the differential equation.

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Answer:

B

Step-by-step explanation:

Domain is the x value and it is asking your for the coordinates with the same x value.

Answer:

The mode is the least used of the measures of central tendency

Step-by-step explanation:

Answer:

Hope this helps!!

Step-by-step explanation:

Answer: i feel like its C but im not sure

Step-by-step explanation:

The standard deviation of the number of companies that are downsizing is approximately 0.817.

The probability of exactly three companies downsizing is approximately 0.275.

To find the average and standard deviation of companies that are downsizing, we use the properties of a binomial distribution. Let's denote the number of companies that are downsizing as a random variable X.

that 13% of companies are downsizing, we have p = 0.13 as the probability of success for each trial (company being downsized) and q = 1 - p = 0.87 as the probability of failure.

The average or expected value of X is given by E(X) = np, where n is the number of trials (sample size). In this case, n = 5.

E(X) = 5 * 0.13 = 0.65

So, the average number of companies that are downsizing is 0.65.

The standard deviation of X is given by σ = √(npq).

σ = √(5 * 0.13 * 0.87) ≈ 0.817

So, the standard deviation of the number of companies that are downsizing is approximately 0.817.

ii. To determine whether it is likely that three companies are downsizing, we can calculate the probability of exactly three successes in five trials using the binomial distribution.

P(X = 3) = (5 choose 3) * 0.13^3 * 0.87^2

P(X = 3) = (5! / (3! * 2!)) * 0.13^3 * 0.87^2

P(X = 3) = 10 * 0.13^3 * 0.87^2 ≈ 0.275

The probability of exactly three companies downsizing is approximately 0.275.

Therefore, it is likely that three companies are downsizing, considering the probability is not extremely low or extremely high.

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Answer:

4x+24

Step-by-step explanation:

To answer this question, first, we will compute the intensity of each earthquake.

Recall that:

Therefore, using the given formula we get:

Then, the intensity of an earthquake that measures 7.5 on the Richter scale is:

and the intensity of an earthquake which measures 2.3 on the Richter scale is:

Now, notice that:

We know that:

Answer:

the length of side OP of rhombus is 45 mm and PM + MN is equal to 90 mm.

We are given a rhombus MNOP

We know that, a rhombus is a quadrilateral in which there are 4 sides and all the sides are equal.

We are given the length of the side NO as:

NO = 45 mm

Now, we need to find the length of side OP.

OP = 45 mm ( as NO = OP)

Also, we need to find PM + MN

PM + MN = NO + NO ( as PM = MN = NO )

= 2 NO

= 2 × 45 mm

= 90 mm

Therefore, we get that, the length of side OP of rhombus is 45 mm and PM + MN is equal to 90 mm.

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Answer:

B

Step-by-step explanation:

Given f(x) then f(x + a) is a horizontal translation of f(x)

• If a > 0 then a shift to the left of a units

• If a < 0 then a shift to the right of a units

Given f(x) then f(x) + c is a vertical translation of f(x)

• If c > 0 then a shift up of c units

• If c < 0 then a shift down of c units

g(x) is f(x) shifted 3 units right and 1 unit up , then

g(x) = (x - 3)² + 1 → B

Answer:

The equation of the blue graph is \(g(x)=(x-3)^{2} +1\). Below is the explanation

Step-by-step explanation:

Given:

   The graph of f(x)=\(x^{2}\)

To find:

    The equation of the transformed graph g(x).

The red graph f(x) is moved right 3 units and up 1 unit to get g(x).

When graph is moved right 3 units , 3 should be subtracted with x.

When graph is moved up 1 unit, 1 is added at the end.

So, our g(x)=\((x-3)^{2} +1\)

The equation of the blue graph is \(g(x)=(x-3)^{2} +1\)

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Answer:

The one on the bottom right

Step-by-step explanation:

Answer:

The last one.

Step-by-step explanation:

Each y goes up by 4: 4,8,12, depending on what the x is. The y is always the x times 4 for the last table.

(I got u homie :P)

Answer:

Option (C) : 3

Step-by-step explanation:

Circumference of front tire = Circumference of back tire

\(3(2\pi\: r) = 2\pi \R\)

\( \R = 3r\)

Answer:

a should be 45

b should be 90

c should also be 45

Step-by-step explanation:

I know this because 180-135=45 for a

since c is also 45, beside c should be 135.

which means those angles are congruent.

since a triangle equals 180 degrees all together 180-45-45=90

so b is 90

correct me if im wrong

The correct statement regarding the probability of choosing a person at random who is a junior and plays video games, and whether these events are independent, is given as follows:

P(junior and video games) = 32%; The two events are not independent.

A probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The outcomes in this problem are given as follows:

Hence the probability is of:

P(junior and video games) = 48/150 = 0.32 = 32%.

The separate probabilities are given as follows:

The multiplication of these probabilities is of:

0.66 x 0.4 = 0.264.

Which is different of 0.32, hence these events are not independent.

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9514 1404 393

Answer:

  B.

Step-by-step explanation:

The symmetric properties of equality and congruence let you swap sides of the equality/congruence symbol without changing the truth of the statement:

  A ≅ B   ⇔   B ≅ A