The decay of 230 mg of an isotope is described by the function A(t)= 230e-0.031t, where t is time in years. Find the amount left after 26 years. Round your answer to the nearest mg.

The measure for <ECA is 115 degree.

When two lines cross at one point, a linear pair of angles is created. If the angles follow the intersection of the two lines in a straight line, they are said to be linear. A linear pair's total angles are always equal to 180°. These are also referred to as additional angles.

Given:

m <ACB = 65 degree

m <ECA = ?

As. <ACB and <ECA forms a linear pair.

So, m<ACB + m<ECA = 180

64 + m <ECA = 180

m <ECA = 180 - 65

m <ECA = 115

Hence, the measure of <ECA = 115 degree.

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Let

x------> the number of men’s shirts

y----->  the number of women’s shirts

we know that

[tex]8x+12y \leq 240[/tex] ----> constraint A    

[tex]x\geq 3[/tex] ----> constraint B

[tex]y\geq 8[/tex] ----> constraint C  

[tex]y\geq 2x[/tex] ----> constraint D  

using a graphing tool

see the attached figure N[tex]1[/tex]

The solution is the shaded area

Remember that

if an ordered pair meets all restrictions and makes sense in the context of the situation, then the ordered pair is a system solution

we're going to proceed to verify each pair ordered

case A) [tex](4,9)[/tex]

Substitute the value of x and y in each  constraint

[tex]x=4\\y=9[/tex]

Verify constraint A

[tex]8*4+12*9 \leq 240[/tex]

[tex]140 \leq 240[/tex] -------> is true

Verify constraint B

[tex]4\geq 3[/tex] -------> is true

Verify constraint C

[tex]9\geq 8[/tex] ------> is true

Verify constraint D

[tex]9\geq 2*4[/tex]

[tex]9\geq 8[/tex] -----> is true

therefore

the ordered pair [tex](4,9)[/tex]  is a system solution

case B) [tex](7,13)[/tex]

Substitute the value of x and y in each  constraint

[tex]x=7\\y=13[/tex]

Verify constraint A      

[tex]8*7+12*13 \leq 240[/tex]

[tex]212 \leq 240[/tex] -------> is true

Verify constraint B

[tex]7\geq 3[/tex] -------> is true

Verify constraint C

[tex]13\geq 8[/tex] ------> is true

Verify constraint D

[tex]13\geq 2*7[/tex]

[tex]13\geq 14[/tex] -----> is not true

therefore

the ordered pair [tex](7,13)[/tex] is not a system solution

case C) [tex](2,15)[/tex]

Substitute the value of x and y in each  constraint

[tex]x=2\\y=15[/tex]

Verify constraint A

[tex]8*2+12*15 \leq 240[/tex]

[tex]196 \leq 240[/tex] -------> is true

Verify constraint B

[tex]2\geq 3[/tex] -------> is not true

Verify constraint C

[tex]15\geq 8[/tex] ------> is true

Verify constraint D

[tex]15\geq 2*2[/tex]

[tex]15\geq 4[/tex] -----> is true

therefore

the ordered pair [tex](2,15)[/tex] is not a system solution  

case D) [tex](5,12)[/tex]

Substitute the value of x and y in each  constraint

[tex]x=5\\y=12[/tex]

Verify constraint A

[tex]8*5+12*12 \leq 240[/tex]

[tex]184 \leq 240[/tex] -------> is true

Verify constraint B

[tex]5\geq 3[/tex] -------> is true

Verify constraint C

[tex]12\geq 8[/tex] ------> is true

Verify constraint D

[tex]12\geq 2*5[/tex]

[tex]12\geq 10[/tex] -----> is true

therefore

the ordered pair [tex](5,12)[/tex] is  a system solution  

case E) [tex](6,15)[/tex]

Substitute the value of x and y in each  constraint

[tex]x=6\\y=15[/tex]

Verify constraint A

[tex]8*6+12*15 \leq 240[/tex]

[tex]228 \leq 240[/tex] -------> is true

Verify constraint B

[tex]6\geq 3[/tex] -------> is true

Verify constraint C

[tex]15\geq 8[/tex] ------> is true

Verify constraint D

[tex]15\geq 2*6[/tex]

[tex]15\geq 12[/tex] -----> is true

therefore

the ordered pair [tex](6,15)[/tex] is  a system solution  

therefore

the answer is

[tex](4,9)[/tex]

[tex](5,12)[/tex]

[tex](6,15)[/tex]

see the attached figure N[tex]2[/tex] to better understand the solution

Answer:

(4,9) (5,12) (6,15)

Step-by-step explanation:

Those were the correct options.

Answer:

option B

Step-by-step explanation:

We have three different function for h(x)

[tex]h(x)= \frac{1}{4}x-4, x<=0[/tex]

WE have x<=0, so the graph starts at x=0 and goes down

Plug in 0 for x

[tex]h(0)= \frac{1}{4}(0)-4[/tex]

So h(0)= -4

When x=0, y= -4

The graph starts at (0,-4) and goes down

[tex]h(x)= \frac{1}{3}x-3, 0<x<=3[/tex]

x lies between 0  and 3, so the graph starts at x=0 and ends at x=3

Plug in 0 for x

[tex]h(0)= \frac{1}{3}(0)-3[/tex]

So h(0)= -3

When x=0, y= -4

Plug in 3 for x

[tex]h(3)= \frac{1}{3}(3)-3=-2[/tex]

When x=3, y= -2

The graph starts at (0,-4) and ends at (3, -2)

[tex]h(x)= \frac{1}{2}x-2, x>=4[/tex]

WE have x>=4, so the graph starts at x=4 and goes to the right

Plug in 4 for x

[tex]h(4)= \frac{1}{2}(4)-2[/tex]

So h(4)= 0

When x=4, y= 0

The graph starts at (4,0) and goes to the right.

SEcond graph is correct

Answer:

B

Step-by-step explanation:

Instantaneous velocity is found by taking the derivative of the position function with respect to time, giving the velocity at a specific moment. Average velocity is calculated by dividing the total displacement by the total time elapsed.

To find instantaneous velocity, we take the derivative of the position function x(t) with respect to time t. This provides the velocity at any given moment. Mathematically, if the position x is a function of time t, then the instantaneous velocity v at time t is defined as:

v = lim_(Δt→0) (x(t + Δt) - x(t)) / Δt

For average velocity, we simply take the total displacement (the change in position) and divide it by the total time elapsed. Suppose we have an object that moves from position x1 to x2 over a time period from t1 to t2, the average velocity ū is:

ū = (x2 - x1) / (t2 - t1)

When the time interval becomes very small, the average velocity approaches the instantaneous velocity.

The composite function that converts temperatures from degrees Kelvin to degrees Fahrenheit is:

K to F = 9/5 * K - 241.15

The image shows a question about how to convert temperatures from degrees Celsius to degrees Fahrenheit using composite functions. The question provides two functions:

[tex]$F(x) = \frac{9}{5}x + 32$[/tex] converts from degrees Celsius to degrees Fahrenheit

[tex]$C(x) = x - 273.15$[/tex]converts from degrees Kelvin to degrees Celsius

We are asked to find the composite function [tex]$C \circ F$[/tex] that converts from degrees Kelvin to degrees Fahrenheit.

To find the composite function, we simply substitute the output of the first function, [tex]$F(x)$[/tex], into the input of the second function, [tex]$C(x)$[/tex]. This gives us:

[tex]```(C \circ F)(x) = C(F(x)) = F(x) - 273.15 = \frac{9}{5}x - 241.15```[/tex]

Therefore, the composite function that can be used to convert temperatures from degrees Kelvin to degrees Fahrenheit is:

[tex]```(C \circ F)(x) = \frac{9}{5}x - 241.15```[/tex]

A composite function is a function that is created by composing two or more other functions. In other words, it is a function that takes the output of one function as the input to another function. Composite functions are often used to solve complex problems that cannot be solved using a single function.

For example, the composite function that we found above,[tex]$(C \circ F)(x)$[/tex], can be used to convert temperatures from degrees Kelvin to degrees Fahrenheit. This is a useful function for scientists and engineers who work with temperatures in different units.

Another example of a composite function is the product rule in calculus. The product rule states that the derivative of the product of two functions [tex]$u(x)$[/tex]and [tex]$v(x)$[/tex] is equal to the sum of the product of the derivatives of [tex]$u$[/tex] and [tex]$v$[/tex] and the product of [tex]$u$[/tex] and the derivative of [tex]$v$[/tex]. This can be expressed mathematically as follows:

[tex]```\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)```[/tex]

The product rule is a composite function because it takes the output of two functions, [tex]$u$[/tex] and [tex]$v$[/tex], and their derivatives,[tex]$u'$[/tex] and [tex]$v'$[/tex], and produces a new function, [tex]$\frac{d}{dx}[u(x)v(x)]$[/tex].

Composite functions are a powerful tool that can be used to solve a wide variety of problems in mathematics, science, and engineering.

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To solve the inequality -138 greater than or equal to -6(6b-7), distribute the -6 to the terms inside the parentheses. Simplify the inequality by subtracting 42 from both sides. Divide both sides by -36, remembering to flip the inequality sign.

To solve the inequality -138 ≥ -6(6b-7), we need to distribute the -6 to the terms inside the parentheses. This gives us -138 ≥ -36b + 42. Next, we can simplify the inequality by subtracting 42 from both sides, resulting in -180 ≥ -36b. To isolate b, we divide both sides of the inequality by -36. Remember when dividing or multiplying an inequality by a negative number, the inequality sign flips. So the final answer is b ≤ 5.

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Answer:

Option 3. 1/14

Step-by-step explanation:

The bag held some coins that are

pennies = 10

nickels = 6

dimes = 5

total coins = 10 + 6 + 5 = 21 coins

You drew 2 coins without replacement,

so the probability that first draw will be nickel = [tex]P_{1}[/tex] = 6/21 = 2/7

Second draw will be a dime  = [tex]P_{2}[/tex] = 5/20 = 1/4

P = [tex]\frac{2}{7}[/tex] × [tex]\frac{1}{4}[/tex] = 2/28 = 1/14

Option 3. 1/14 would be the answer.

Final answer:

The probability of withdrawing a nickel and then a dime from the bag without replacement is 1/14, calculated by multiplying the independent probabilities of drawing each coin in sequence (6/21 for the nickel and 5/20 for the dime).

Explanation:

The student has asked for the probability of withdrawing first a nickel and then a dime from a bag containing ten pennies, six nickels, and five dimes, without replacement. To solve this, we calculate the probability of each event independently and then multiply these probabilities, since the events are dependent.

First, the probability of drawing a nickel: There are six nickels and a total of 21 coins, so the probability is 6/21. After drawing a nickel, there are five dimes left out of 20 coins total, thus the second probability is 5/20. Multiplying these together gives us:

(6/21) * (5/20) = 30/420 = 1/14

So, the probability of drawing a nickel and then a dime without replacement from this bag of coins is 1/14.

The equation 2m - nx = x + 4 is a linear equation solvable for x using algebraic manipulation. The solution is x = (2m - 4)/(nx + 1), requiring values for m and n to find x's specific value.

The given equation 2m - nx = x + 4 is a linear equation and can be solved for x by applying algebraic techniques. The question indicates that there is an element of differential equations involved, suggesting that it might be related to more advanced topics such as linear differential equations or characteristics equations. However, the provided context also mentions concepts like probability with positive and negative jumps, quadratic equations, and principles from physics, but these do not directly apply to the initial linear equation provided.

To solve for x in the given equation, we would rearrange the terms to isolate x:

2m - nx = x + 4

2m - 4 = x(nx + 1)

x = (2m - 4)/(nx + 1)

The solution for x is now expressed in terms of m and n. To find the specific value of x, values for m and n must be known.

Answer: 1773.54 mg

Step-by-step explanation:

Given: The daily dose of ampicillin for the treatment of an ear infection is 115 mg/kg of body weight.

The weight of a child = 34 lb

We know that 1 pound = 0.453592 kg

Then , the weight of child in kilograms = [tex]0.453592\times34=15.422128\text{ kg}[/tex]

Now, the daily dose of ampicillin for the treatment of an ear infection for the child is given by :-

[tex]115\times15.422128=1773.54472\approx1773.54\text{ mg}[/tex]

Hence, the  daily dose for a 34 lb child = 1773.54 mg

The conversion rate from Ib(pounds) to kg(kilograms)

1 Ib = 0.453592kg

Step 1

We convert the weight of the child to from lb to kg

1 Ib = 0.453592kg

34 lb = ?

Cross Multiply

34 x 0.453592kg

= 15.4221 kg

The weight of the child in kg = 15.4221 kg

Step 2

The daily dose of ampicillin for the treatment of an ear infection is 115 mg/kg of body weight.

Therefore the daily dose of ampicillin for the treatment of ear infection for the child = 15.4221 kg x 115 mg/kg

= 1,773.5415 mg

The daily dose for a 34 lb child is approximately to nearest whole number is 1,774 mg

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The employer would need to pay $19.13 in payroll taxes for this employee.

Payroll tax is a tax that employers are required to deduct from their employee's wages and pay to the government on their behalf.

It is a tax on the wages or salaries that employees earn and is used to fund various government programs, such as Social Security, Medicare, and unemployment insurance.

We have,

The amount an employer needs to pay for payroll taxes depends on several factors.

The employee's gross wages

The type of payroll taxes.

The tax rates are in effect.

Now,

The two primary types of payroll taxes that employers are responsible for are:

Social Security tax:

This tax funds the Social Security program and is calculated as a percentage of the employee's gross wages, up to a certain limit.

In 2023, the Social Security tax rate is 6.2% for both employers and employees, and the wage base limit is $147,000.

Medicare tax:

This tax funds the Medicare program and is calculated as a percentage of the employee's gross wages, with no limit.

In 2023, the Medicare tax rate is 1.45% for both employers and employees.

Assuming the employee owes $250 for Social Security and Medicare taxes combined, and the employer is responsible for paying the same percentage rate as the employee,

The employer would need to pay an additional $250 in payroll taxes.

i.e

Social Security tax.

= $250 x 6.2%

= 250 x 6.2/100

= $15.50

Medicare tax.

= $250 x 1.45%

= 250 x 1.45/100

= $3.63

Total payroll taxes owed by the employer.

= $15.50 + $3.63

= $19.13

Therefore,

The employer would need to pay $19.13 in payroll taxes for this employee.

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Final answer:

68% of the total combined amount of paint is used for the job, calculated by using 20% of the blue paint and 80% of the crimson paint from the initial amounts of 2 liters and 8 liters, respectively.

Explanation:

To calculate the percentage of the total combined amount of paint used during the job, we first identify the amount of blue and crimson paint used separately. From the question we know that there were initially 2 liters of blue paint and 8 liters of crimson paint.

Now, we compute the total paint used and the initial total paint:

To find the percentage of the total paint used, we divide the total paint used by the initial total paint and multiply by 100:

Percentage used = (Total paint used / Initial total paint) × 100

Percentage used = (6.8 liters / 10 liters) × 100 = 68%

Therefore, 68% of the total combined amount of paint is used for the job.

Answer:

Step-by-step explanation:

You want the reasons for each of the final three steps in the solution of the inequality 7x +16 < -5(5 -5x).

2. 7x +16 < -20 +25x . . . . . . . given step 2

3.  16 < -20 +18x . . . . . Subtraction property of inequality

4.  36 < 18x . . . . . . . . . Addition property of inequality

5.  2 < x . . . . . . . . . . . . Division property of inequality

__

Additional comment

Subtraction of 7x is the same as adding -7x, so you can say the reason for step 3 is either of "subtraction property" or "addition property" The same would be true of step 4.

To figure this out, you compare the inequality on each line with the one before, and determine what was changed. If the same thing is done to both sides of the inequality (addition, subtraction, multiplication, division), the reason is the corresponding property of inequality.

The multiplication and division properties of inequality tell you to reverse the comparison symbol when the factor used is negative. Here, the factor is 18, a positive number, so the inequality symbol remains untouched.

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