Find the point of intersection of the pair of straight lines.

10x - 4y = 43

-3x - 3y = -15

(x, y) = ( , )

Answer:

2 mg/minute.

Step-by-step explanation:

We have been given that a drug is mixed 2 g in 500 ml.

First of all, we will convert 2 grams to milligrams. 1 gram equals 1000 milligrams.

2 grams = 2,000 mg.

Now, we will find amount of mg per ml as:

[tex]\text{The amount of mg per ml}=\frac{2000\text{ mg}}{\text{500 ml}}[/tex]

[tex]\text{The amount of mg per ml}=\frac{4\text{ mg}}{\text{ml}}[/tex]

We have been given that a lidocaine drip is infusing at 30 mL/hr on an infusion device, so amount of mg per hr would be:

[tex]\text{Amount of mg per hour}=\frac{30\text{ ml}}{\text{ hr}}\times \frac{4\text{ mg}}{\text{ml}}[/tex]

[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{ hr}}[/tex]

We know 1 hour equals 60 minutes.

[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{ hr}}\times \frac{\text{1 hour}}{\text{60 minutes}}[/tex]

[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{60 minutes}}[/tex]

[tex]\text{Amount of mg per hour}=\frac{2\text{ mg}}{\text{minute}}[/tex]

Therefore, the patient is receiving 2 mg/minute.

To calculate the mg/minute that the patient is receiving from the lidocaine drip, we need to convert the given flow rate from mL/hr to mL/minute, and then convert the drug concentration from grams to milligrams.

To calculate the mg/minute that the patient is receiving, we need to convert the given flow rate from mL/hr to mL/minute, and then convert the drug concentration from grams to milligrams. Here's how:

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Answer:

a) Since the sphere intersects the xy-plane then the set of points of the sphere nearest to the xy-plane is the set of points in the circumference [tex](x+5)^2+y^2=18[/tex].

b)(-14.9, 0, 9 )

Step-by-step explanation:

a) The centre of the sphere is (-5,0,-9) and the radio of the sphere is [tex]\sqrt{99} \sim 9.9[/tex]. Since |-9|=9 < 9.9,  then the sphere intersect the xy-plane and the intersection is a circumference.

Let's find the equation of the circumference.

The equation of the xy-plane is z=0. Replacing this in the equation of the sphere we have:

[tex](x+5)^2+y^2+9^2=99[/tex], then [tex](x+5)^2+y^2=18[/tex].

b) Observe that the point (-9,0,9) has the same y and z coordinates as the centre and the x coordinate of the point is smaller than that of the x coordinate of  the centre. Then the point of the sphere nearest to the given point will be at a distance of one radius from the centre, in the negative x direction.

(-5-[tex]\sqrt{99}[/tex], 0, 9)= (-14.9, 0, 9 )

Answer:

Step-by-step explanation:

All the right triangles are similar, so all will have the same angles.

The missing angle (B) in ΔABC is the complement of the given one:

  ∠DBC = 90° - 65° = 25°

The missing angles in the smaller triangles are the complements of the known acute angles in those triangles.

A diagram can help you see this.

Answer:

$ 4933.2 ( approx )

Step-by-step explanation:

∵ Future value formula is,

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]

Where,

P = principal amount,

r = annual rate,

n = number of periods,

t = number of years,

Given,

P = $ 4,000, r = 3.5 % = 0.035, t = 6 years n = 12 ( number of months in 1 year = 12 ),

Hence, the future value would be,

[tex]A=4000(1+\frac{0.035}{12})^{72}=4933.20414683\approx \$ 4933.2[/tex]

Answer:

This isn't true.

Step-by-step explanation:

Think of the case a=2, b=3 and c=6. We have that a|b, since 2|6.

We also have that ab=c, since 2*3=6. However, it is NOT true that a|b, as 2 does NOT divide 3. As this you can construct many other examples where a|c and ab=c BUT a does NOT divide b.

Other counterexamples:

a=2, b=5, c=10

a=2, b=7, c=14

a=2, b=11, c=22

a=2, b=13, c=26

Answer:

P= -2x +46

Step-by-step explanation:

the relation between price and demand is

P= mx +b ........................1

when demand is 10 shirts price is $26

when demand is 20 shirts price is $6

firstly put P= 26 and x= 10 in 1

26= 10m + b.......................2

secondly put x= 20 and P= 6 in 1

6= 20m + b ............................3

solving  2 and 3 we get

m = -2

putting this value of m in either of 2 and 3 to get b

b= 46

so the final relation obtained by putting m= -2 and b= 46 in 1 we get

P= -2x +46

Answer:

Given dose (i.e 600 mg) lies outside the recommended range of  271.8-543.6 mg/day that too on the higher side

Hence, the prescription is not safe.

Step-by-step explanation:

Given:

Weight of medicine given per interval = 150 mg

time interval = 6 hours

thus, number of intervals per day = [tex]\frac{\textup{24}}{\textup{6}}[/tex] = 4

therefore,

the total dose of medicine provided per day = 4 × 150 = 600 mg

Now,

Recommended dosage =  20-40 mg/kg/day

weight of child = 30 lb

also,

1 lb = 0.453 kg

thus,

weight of child = 30 × 0.453 = 13.59 kg

Therefore, the recommended dose for the child

=  (  20-40 mg/kg/day ) × 13.59

= 271.8-543.6 mg/day

now,

the given dose (i.e 600 mg) lies outside the recommended range of  271.8-543.6 mg/day that too on the higher side

Hence, the prescription is not safe.

Answer:

The set of solutions is [tex]\{\left[\begin{array}{c}a\\b\\x\\y\\z\end{array}\right] = \left[\begin{array}{c}-26+503y+543z\\-37+655y+724z\\-4+80y+90z\\y\\z\end{array}\right] : \text{y, z are real numbers}\}[/tex]

Step-by-step explanation:

The matrix associated to the problem is [tex]A=\left[\begin{array}{ccccc}1&-1&2&-8&1\\2&-1&-4&1&-2\\-4&1&4&-3&-1\end{array}\right][/tex] and the vector of independent terms is (3,1,-1)^t. Then the matrix equation form of the system is Ax=b.

The vector equation form is [tex]a\left[\begin{array}{c}1\\2\\-4\end{array}\right]+b\left[\begin{array}{c}-1\\-1\\1\end{array}\right] + x\left[\begin{array}{c}2\\-4\\4\end{array}\right]+y\left[\begin{array}{c}-8\\1\\-3\end{array}\right] + z\left[\begin{array}{c}1\\-2\\-1\end{array}\right]=\left[\begin{array}{c}3\\1\\-1\end{array}\right][/tex].

Now we solve the system.

The aumented matrix of the system is [tex]\left[\begin{array}{cccccc}1&-1&2&-8&1&3\\2&-1&-4&1&-2&1\\-4&1&4&-3&-1&-1\end{array}\right][/tex].

Applying rows operations we obtain a echelon form of the matrix, that is [tex]\left[\begin{array}{cccccc}1&-1&2&-8&1&3\\0&1&-8&-15&-4&-5\\0&0&1&-80&-9&-4\end{array}\right][/tex]

Now we solve for the unknown variables:

Since the system has two free variables then has infinite solutions.

The set of solutions is [tex]\{\left[\begin{array}{c}a\\b\\x\\y\\z\end{array}\right] = \left[\begin{array}{c}-26+503y+543z\\-37+655y+724z\\-4+80y+90z\\y\\z\end{array}\right] : \text{y, z are real numbers}\}[/tex]

Answer:

a person should consume 1340 calories after eating ice cream

Step-by-step explanation:

Given :

The worst treat tested by the group was 1,910 total calories.

People need roughly 3,100 to 3,400 calories per day.

To Find : Using a daily average, how many additional calories should a person consume after eating ice cream?

Solution:

People need roughly 3,100 to 3,400 calories per day.

So, Average = [tex]\frac{3100+3400}{2}[/tex]

Average = [tex]3250[/tex]

So, one should take 3250 calories daily

The worst treat tested by the group was 1,910 total calories.

So, additional calories should a person consume after eating ice cream :

= 3250-1910

= 1340

Hence a person should consume 1340 calories after eating ice cream

Answer:

1. g(x)=2x+1-3 --> g(x)=2x-2, which is also y=2x-2, so you can graph it.

Step-by-step explanation:

Question 1: If f(x) = 2x+1, then you can see that all you have to do is substitute the equation for f(x) into the g(x) equation because g(x)= f(x)-3. So, if you substitute it, the equation will be g(x) = (2x+1) -3, then you just solve the rest of the equation. Put it into slope intercept form, y=mx+b, and then graph the equation.

Sorry, I don't really understand number 2 myself, so hopefully I could help with he first one.

Final answer:

To solve the system of inequalities, first, solve each inequality separately. Then, combine the solutions to find the common range of values for x that satisfy all the inequalities.

Explanation:

To solve the system of inequalities:

2x - 1 < x + 3

5x - 1 > 6 - 2x

x - 5 < 0

So, the solution to the system of inequalities is: x < 4, x > 1, x < 5

Answer:

Not always we can use a calculator to determine if a number is rational or irrational.

Step-by-step explanation:

Consider the provided information.

Can you ever use a calculator to determine if a number is rational or irrational.

Irrational   number: A   number is irrational if it cannot   be   expressed by dividing two     integers. The decimal expansion of     Irrational numbers are neither terminate nor     periodic.

The calculators gives the approximate answer, whether the number is irrational or rational.

So, it would be difficult to identify whether a large number produced by the calculator is irrational or not. As we know that many rational numbers can be incredibly large.

So, we can say that not always we can use calculator to determine if a number is rational or irrational.

Thus, Not always we can use a calculator to determine if a number is rational or irrational.

Answer:

3.055 m

Step-by-step explanation:

In this solution we will use next notation:

[tex]t_1[/tex]=  time elapsed since oco serves the ball until it reaches its opponent.

[tex]t_2[/tex]=  time elapsed since the opponent returns the ball until it reaches oco.

d= Total distance traveled by Oco since serving the ball until  meeting the return.

We know that oco serves at vs = 50 m/s and her opponent is x=25 m away. Then, t_1 is given by

[tex]t_1=\frac{25m}{50m/s}=0.5s[/tex]

To compute t_2 observe that the return speed is 12.5 m/s and the distance that the ball will travel is [tex]25-(10t_1+10t_2)[/tex]. Then,

[tex]t_2=\frac{25-10t_1-10t_2}{12.5}=\frac{20-10t_2}{12.5}\implies t_2=\frac{20}{22.5}=\frac{8}{9}s[/tex].

Therefore,

[tex]d=10(t_1+t_2)=10(0.5+\frac{8}{9})=10(\frac{17}{18})=\frac{85}{9}m[/tex]

Finally, as Oco started 12.5m away from the net, when she meets the return she will be

[tex]12.5-\frac{85}{9}=\frac{55}{18}=3.055m[/tex]

away from the net.

Answer:

2x² + 4x + 3 = 0

Step-by-step explanation:

The function is said to be quadratic if it has highest degree = 2.

Further, The standard form of Quadratic Equation is:

ax² + bx + c = 0

where, a ≠ 0

a, b and c are constants

and x is unknown variable.

Thus, The Standard form of given Quadratic Equation is 2x² + 4x + 3 = 0

To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, complete the square to get f(x) = 2(x + 1)^2 - 5, which reveals the vertex of the parabola at (-1, -5).

To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, also known as vertex form, we need to complete the square. The standard form of a quadratic function is typically written as f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola formed by the graph of the quadratic equation.

Here's how we can rewrite the given quadratic function:

In this form, it is clear that the vertex of the parabola is at (-1, -5).

By using the principles of linear interpolation, the yield of strawberries with 75 cubic feet of fertilizer can be calculated as approximately 616.25 pounds.

The yield of strawberries based on the amount of fertilizer fed to the plants can be estimated using linear interpolation. We can establish two points based on the given information: (70, 770) and (100, 1100), where the first number represents the amount of fertilizer and the second one, the yield. The interpolation line equation can be formulated as y = mx + c where m = (y2 - y1) / (x2 - x1); as such, m = (1100 - 770) / (100 - 70) = 8.25.

To find the value of c (y-intercept), we use the equation with one of the known points and solve c = y1 - m * x1 = 770 - 8.25 * 70 = -5.

The yield, y at 75 cubic feet of fertilizer can be calculated as y = 8.25 * 75 - 5 = 616.25. Therefore, the estimated yield of strawberries when 75 cubic feet of fertilizer is applied is approximately 616.25 pounds.

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Answer:

C

Step-by-step explanation:

A can be from 2- 9 ( 8 digits)

B can be 0 to 9 (10 digits)

C can be 0 to 9 (10 digits)

Each of the X's can be 0 to 9 (10 digits)

To get the number of possibilities, we multiply them to get:

8 * 10 * 10 * 10 * 10 * 10 * 10 = 8,000,000

But now, 1 number (867-5309) is restricted, so the number of possibilities decrease by 1:

8,000,000 - 1= 7, 999, 999

Correct answer is C

Answer:

0.9796

Step-by-step explanation:

Given that 88%of bolts and 83% of nails meet specifications.

This implies that for a randomly selected bowl the prob that it meets specifications = P(A) = 0.88

Similarly, for a randomly selected bolt, it meets specifications is P(B) = 0.83

We know that bolt and nail are independent of each other.

Hence [tex]P(A \bigcap B) = P(A)P(B)\\\\=0.88*0.83=0.7304[/tex]

Required probability = Probability that atleast one of them meets specifications)

[tex]= P(AUB)\\=P(A)+P(B)-P(A \bigcap B)\\=0.88+0.83-0.7304\\=1.71-0.7304\\=0.9796[/tex]

Final answer:

To find the probability that at least one of the chosen bolt or nail meets specifications, use the complement rule. Multiply the probabilities that each does not meet specifications and subtract from 1 to find the probability that at least one meets specifications.

Explanation:

To find the probability that at least one of the chosen bolt or nail meets specifications, we can use the complement rule. The complement of the event that at least one meets specifications is the event that none of them meet specifications. The probability that the bolt does not meet specifications is 1 - 0.88 = 0.12, and the probability that the nail does not meet specifications is 1 - 0.83 = 0.17. Since the events are chosen independently, we can multiply these probabilities together to get the probability that both do not meet specifications: 0.12 x 0.17 = 0.0204.

Using the complement rule, we subtract this probability from 1 to find the probability that at least one meets specifications: 1 - 0.0204 = 0.9796. Therefore, the probability that at least one of the bolt and nail meets specifications is approximately 0.9796.

Answer:

Step-by-step explanation:

Given that,

Ken has 7 base ball

And each base ball have a weight of 0.3 pounds

If 1 ball = 0.3 pounds

Then, 7 baseball = 7 × 0.3 pounds

So, 7 baseball = 2.1 pounds

Then, the total weight of the 7base balls is 2.1 pounds

But, we want to to use the box to represent this data following the given information in the attachment

Note: since, the weight of the baseball is in decimal points

Let 0.1 pounds be 1 Square box.

Answer

1. To represent the weight of one base ball (0.3 pounds), ken should shade 3 Square box.

Since 1 pounds is 1 Square box

2. To represent the weight of all the seven base ball (2.1pounds), he should shade this amount seven times.

This means that he need to shade 3 square box seven times. This shows that the amount used in the second part of the question means by how much must the square be shaded compare to question the first part of the question

3. The shaded part of the model will represent the expression 0.3 × 7

4. The total weight of the base ball is 2.1 pounds.

Given that ;

Ken has no. of baseballs = 7

And each baseball have a weight is = 0.3 pounds

By unity method;

If 1 ball = 0.3 pounds

Then, 7 baseball = 7 × 0.3 pounds

So, 7 baseball = 2.1 pounds

Then, the total weight of the 7 baseballs is 2.1 pounds.

As per given in the question ;

We want to use the box to represent this data given in the following question .  

Since, the weight of the baseball is in decimal points

Let 0.1 pounds be 1 Square box.

Since 1 pounds is 1 Square box.

This means that he need to shade 3 square box seven times. This shows that the amount used in the second part of the question means by how much must the square be shaded compare to question .  

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Answer:

  The product is approximately 5400.

Step-by-step explanation:

  9 1/16 rounded to the nearest whole number is 9.

  645 3/5 rounded to the nearest hundred is 600.

Multiplying these values gives a product of 9·600 = 5400.

The product is approximately 5400.

Answer:

5,400.

Step-by-step explanation:

9 1/6 = 9 to the nearest whole number.

645 3/5 = 600 to the nearest hundred.

The produce = 9 * 600 = 5400.

Answer:

The correct answer is B. : 10 sq mm = 0.1 sq cm

Step-by-step explanation:

It is just a matter of changing the units. The equivalence we need to know is 1cm = 10 mm. Also, we need to have in mind that we can write 10 sq mm as 10 mm*mm, because : 10 sq mm = 10 mm² = 10 mm*mm

Now we multiply two times by the fraction (1cm / 10 mm), which does not alter our measurement because the fraction is the same as multiplying by 1.

10 sq mm = 10 mm* mm = (10 mm*mm)*(1 cm / 10 mm)*(1 cm / 10 mm) = (10 mm*mm*cm*cm/ 10*10 mm*mm) =10/100 cm*cm = 0.1 cm² = 0.1 sq cm

Therefore, we have the equivalency : 10 sq mm = 0.1 sq cm

The 99% confidence interval estimate for the genuine proportion of families who own at least one DVD player is:

Lower bound: 0.287.

Upper bound: 0.563.

To find the 99% confidence interval estimate of the true proportion of families who own at least one DVD player, follow these steps:

Step 1. Determine the sample proportion [tex](\( \hat{p} \))[/tex]:

Number of families surveyed ( n ) = 85

Number of families owning at least one DVD player ( x ) = 36

Sample proportion [tex](\( \hat{p} \)) = \( \frac{x}{n} = \frac{36}{85} = 0.4247 \)[/tex]

Step 2. Find the standard error (SE) of the sample proportion:

Standard error formula: [tex]\( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)[/tex]

Plug in the values: [tex]\( SE = \sqrt{\frac{0.4247 \times (1 - 0.4247)}{85}} = \sqrt{\frac{0.4247 \times 0.5753}{85}} = \sqrt{\frac{0.2443}{85}} = \sqrt{0.002875} = 0.0536 \)[/tex]

Step 3. Determine the z-value for a 99% confidence interval:

The z-value for a 99% confidence interval is approximately 2.576.

Step 4. Calculate the margin of error (ME):

Margin of error formula: [tex]\( ME = z \times SE \)[/tex]

Plug in the values: [tex]\( ME = 2.576 \times 0.0536 = 0.1381 \)[/tex]

Step 5. Determine the confidence interval

Lower limit: [tex]\( \hat{p} - ME = 0.4247 - 0.1381 = 0.2866 \)[/tex]

Upper limit: [tex]\( \hat{p} + ME = 0.4247 + 0.1381 = 0.5628 \)[/tex]

Therefore, the 99% confidence interval estimate of the true proportion of families who own at least one DVD player is:

Lower limit: 0.287

Upper limit: 0.563

Complete Question:

A survey of 85 families showed that 36 owned at least one DVD player. Find the 99 \% confidence interval estimate of the true proportion of families who own at least on DVD player. Place your limits, rounded to 3 decimal places, in the blanks. Do not use any labels or symbols other than the decimal point. Simply provide the numerical values. For example, 0.123 would be a legitimate entry.

Lower limit (first blank) [tex]$=$ $\qquad$[/tex] ______ , Upper limit (second blank) = _______

Answer: Hi!

First, if you think that a compass has degrees as units, then N50E would be

50 degrees from north in the direction of the east, so if you put our 0 in east and count counterclockwise this will be an angle of 40 degrees.

If you think north has te Y axis positive direction, and east as the X axis positive direction. then the first boat has an angle of 40° counterclockwise from the +x

so the velocity in y is Vy=30mph*sin(40°) and in x is Vx= 30mph*cos(40°)

then the total displacement will be 22.98m to east and 19.28 north

the second one goes to s 70 e, so using the same notation as before, you can write this has -20° degrees count counterclockwise.

so decomposing the velocity will give us

Vy = 26*sin(-20°) and the displacement in Y is -8.89m

Vx = 26*cos(-20°) and the displacement in X is 24.43m

so the distance between the boats in y will be 19.28m - (-8.99)m = 28.27m

and in x: 24.43m - 22.98m = 1.45m

and the total distance is [tex]D^{2} = 1.45^{2} + 28.27^{2}[/tex]

so D = 28.30 m

Answer:

The statement [tex]((p \lor q) \land (p \implies r) \land (\neg r)) \implies q[/tex] is a tautology.

Step-by-step explanation:

A truth table shows how the truth or falsity of a compound statement depends on the truth or falsity of the simple statements from which it's constructed.

We can see from the truth table that the last column contains only true values. Therefore, the statement is a tautology.

Logical equivalences are a type of relationship between two statements or sentences in propositional logic. To simplify an equivalency, start with one side of the equation and attempt to replace sections of it with equivalent expressions. Continue doing this until you have achieved the desired statement form.

[tex]((p \lor q) \land (p \implies r) \land (\neg r)) \implies q \\\equiv \neg[(p \lor q) \land (p \implies r) \land (\neg r)] \lor q[/tex] by implication law

[tex]\equiv \neg[(p \lor q) \land (\neg p \lor r) \land (\neg r))] \lor q[/tex] by implication law

[tex]\equiv \neg(p \lor q) \lor \neg (\neg p \lor r) \lor \neg(\neg r) \lor q[/tex] by de Morgan’s law

[tex]\equiv \neg(p \lor q) \lor \neg (\neg p \lor r) \lor r \lor q[/tex] by Double Negative

[tex]\equiv [(\neg p \land \neg q) \lor (p \land \neg r)] \lor r \lor q[/tex] by de Morgan’s law

[tex]\equiv [(\neg p \land \neg q) \lor q] \lor [(p \land \neg r) \lor r][/tex] by commutative and associative laws

[tex]\equiv [(\neg p \lor q) \land (\neg q \lor q)] \lor [(p \lor r) \land (\neg r \lor r)][/tex] by distributive laws

[tex]\equiv (\neg p \lor q) \lor (p \lor r)[/tex] by negation and identity laws

[tex]\equiv (\neg p \lor p) \lor (q \lor r)[/tex] by communicative and associative laws

[tex]\equiv T[/tex] by negation and domination laws

Therefore, the statement is a tautology.

The given logical statement is a tautology, as confirmed by a truth table and verified through logical equivalences, specifically equivalent to "q OR ~q," demonstrating its truth in all possible scenarios.

To determine whether the given logical statement "((p OR q) AND (p -> r) AND (~r)) -> q" is a contradiction, a tautology, or neither, we can create a truth table. The statement has three propositional variables: p, q, and r, so we need a truth table with 2^3 = 8 rows to cover all possible combinations of truth values for these variables.

p | q | r | (p OR q) | (p -> r) | (~r) | ((p OR q) AND (p -> r) AND (~r)) | (((p OR q) AND (p -> r) AND (~r)) -> q)

--|---|---|----------|----------|-----|---------------------------------|-----------------------------------------

T | T | T | T        | T        | F   | F                               | T

T | T | F | T        | F        | T   | F                               | T

T | F | T | T        | T        | F   | F                               | T

T | F | F | T        | F        | T   | F                               | T

F | T | T | T        | T        | F   | F                               | T

F | T | F | T        | T        | T   | T                               | T

F | F | T | F        | T        | F   | F                               | T

F | F | F | F        | T        | T   | F                               | T

In the last column, we evaluate the given logical statement "((p OR q) AND (p -> r) AND (~r)) -> q" for each row.

Now, let's analyze the results:

- The statement is True in all rows. Therefore, it is a tautology because it is always true, regardless of the truth values of p, q, and r.

We can also verify this using logical equivalences. The statement "((p OR q) AND (p -> r) AND (~r)) -> q" is logically equivalent to "q OR ~q," which is always true by the law of excluded middle. This confirms that the original statement is a tautology.

To learn more about tautology

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Answer:  The required answer is

[tex]f(x)=e^x+\cos x+\sin x.[/tex]

Step-by-step explanation:  We are given to find the inverse Laplace transform of the following function as a function of x :

[tex]F(s)=\dfrac{2s^2}{(s-1)(s^2+1)}.[/tex]

We will be using the following formulas of inverse Laplace transform :

[tex](i)~L^{-1}\{\dfrac{1}{s-a}\}=e^{ax},\\\\\\(ii)~L^{-1}\{\dfrac{s}{s^2+a^2}\}=\cos ax,\\\\\\(iii)~L^{-1}\{\dfrac{1}{s^2+a^2}\}=\dfrac{1}{a}\sin ax.[/tex]

By partial fractions, we have

[tex]\dfrac{s^2}{(s-1)(s^2+1)}=\dfrac{A}{s-1}+\dfrac{Bs+C}{s^2+1},[/tex]

where A, B and C are constants.

Multiplying both sides of the above equation by the denominator of the left hand side, we get

[tex]2s^2=A(s^2+1)+(Bs+C)(s-1).[/tex]

If s = 1, we get

[tex]2\times 1=A(1+1)\\\\\Rightarrow A=1.[/tex]

Also,

[tex]2s^2=A(s^2+1)+(Bs^2-Bs+Cs-C)\\\\\Rightarrow 2s^2=(A+B)s^2+(-B+C)s+(A-C).[/tex]

Comparing the coefficients of x² and 1, we get

[tex]A+B=2\\\\\Rightarrow B=2-1=1,\\\\\\A-C=0\\\\\Rightarrow C=A=1.[/tex]

So, we can write

[tex]\dfrac{2s^2}{(s-1)(s^2+1)}=\dfrac{1}{s-1}+\dfrac{s+1}{s^2+1}\\\\\\\Rightarrow \dfrac{2s^2}{(s-1)(s^2+1)}=\dfrac{1}{s-1}+\dfrac{s}{s^2+1}+\dfrac{1}{s^2+1}.[/tex]

Taking inverse Laplace transform on both sides of the above, we get

[tex]L^{-1}\{\dfrac{2s^2}{(s-1)(s^2+1)}\}=L^{-1}\{\dfrac{1}{s-1}\}+L^{-1}\{\dfrac{s}{s^2+1}+\dfrac{1}{s^2+1}\}\\\\\\\Rightarrow f(x)=e^{1\times x}+\cos (1\times x)+\dfrac{1}{1}\sin(1\times x)\\\\\\\Rightarrow f(x)=e^x+\cos x+\sin x.[/tex]

Thus, the required answer is

[tex]f(x)=e^x+\cos x+\sin x.[/tex]

Answer:

In order to have a consistent linear system represented by the augmented matrix:

[tex]\left[\begin{array}{ccc}2&-3&h\\-6&9&5\end{array}\right][/tex]

the value of h must be:

[tex]h=-\frac{5}{3}[/tex]

Step-by-step explanation:

A system is consistent if it has a solution, this solution can be unique or a set of infinite solutions.  

First, you take the augmented matrix and find the equivalent row echelon form using Gaussian-Jordan elimination:

To do this, you have to multiply the 1st row by 3 and add it to the 2nd row, the resulting matrix is:

[tex]\left[\begin{array}{ccc}2&-3&h\\0&0&5+3h\end{array}\right][/tex]

Now, write the system of equations:

[tex]2x_1-3x_2=h\\0x_1+0x_2=5+3h[/tex]

The only way this system has a solution is if 5+3h=0, then, to satisfy this, the value of h must be:

[tex]h=-\frac{5}{3}[/tex]

Answer:

D

Step-by-step explanation:

Because ordering in ratios is important, so it must stay constant like 6,15.

Answer:

The answer is: D) 15/6

Step-by-step explanation:

The ratio of two given numbers such as X and Y is expressed by the symbol ':' Therefore, the ratio of X and Y or X:Y can be referred to as X is to Y and can also be expressed as a fraction X/Y or X÷Y.

Therefore, the ratio can be expressed in a number of ways, 6:15 = 6 to 15 = 6/15

Whereas, 15/6 = 15:6 ≠ 6:15

Answer:

Her weight is increase by 18 lbs over past five years and the slope is 3.6 lbs per year.

Step-by-step explanation:

Given information: Estelle weight is

At age 16 =  110 lbs

At age 21 = 128 ibs

Increase in her weight over the past 5 years is the difference of weight at age 21 and at age 16.

Increase in her weight over the past 5 years = 128 - 110 = 18

Her weight is increase by 18 lbs over past five years.

Let x=age and y=weight, then the weight function passes through the points (16,110) and (21,128).

If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex],  then the slope of the line is

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Using the above formula we get

[tex]m=\frac{128-110}{21-16}[/tex]

[tex]m=\frac{18}{5}[/tex]

[tex]m=3.6[/tex]

Therefore the slope is 3.6 lbs per year.

Answer:

30 micro grams

Step-by-step explanation:

1 ml contains 0.05 mg (milligram) of digoxin

So, 0.6 ml contains digoxin = [tex]0.6 \times 0.05[/tex]

                                          = [tex]0.03 mg[/tex]

Now 1 mg contains 1000 μg (micro grams)

So, 0.03 mg contains micro grams= [tex]0.03 \times 1000[/tex]

                                                      = [tex]30[/tex]

Hence 30 micro grams of digoxin would be delivered in each dose of 0.6 ml .

Answer:

(a) The probability is 9.49%

(b) The probability is 38.28%

Step-by-step explanation:

The probability that a failure is due to induced substance is calculated as a multiplication as:

(13%) * (73%) = 9.49%

Where 13% is the percentage of heart failures that are due outside factors and 73% is the percentage of outside factors that are due induced substances.

On the other hand, the probability that a failure is due to disease or infection is the sum of the probability that a failure is due to disease and the probability that a failure is due to infection.

Then, the probability that a failure is due to disease is calculated as:

(87%) * (27%) =  23.49%

Where 87% is the percentage of heart failures that are due natural factors and 27% is the percentage of natural factors that are due disease.

At the same way, the probability that a failure is due to infection is calculated as:

(87%) * (17%) =  14.79%

So, the probability that a failure is due to disease or infection is:

23.49% + 14.79% = 38.28%

Answer:  [tex]\sin(x+y)=\dfrac{1-8\sqrt{3}}{15}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\sin x=\dfrac{1}{3}\\\\so,\\\\\cos x=\sqrt{1-\dfrac{1}{9}}=\sqrt{\dfrac{8}{9}}=\dfrac{2\sqrt{2}}{3}[/tex]

Since x ends in the 2 nd quadrant,

So, [tex]\cos x=\dfrac{-2\sqrt{2}}{3}[/tex]

Similarly,

[tex]\cos y=\dfrac{1}{5}\\\\So,\\\\\sin y=\sqrt{1-\dfrac{1}{25}}=\sqrt{\dfrac{24}{25}}=\dfrac{2\sqrt{6}}{5}[/tex]

So, sin(x+y) is given by

[tex]\sin x\cos y+\sin y\cos x\\\\\\=\dfrac{1}{3}\times \dfrac{1}{5}+\dfrac{2\sqrt{6}}{5}\times (-)\dfrac{2\sqrt{2}}{3}\\\\\\=\dfrac{1}{15}-\dfrac{8\sqrt{3}}{15}\\\\\\=\dfrac{1-8\sqrt{3}}{15}[/tex]

Hence, [tex]\sin(x+y)=\dfrac{1-8\sqrt{3}}{15}[/tex]