Use addition to solve the linear system of equations. Include all of your work in your final answer.

x - y = 4
x+ 2y = 4

The distance traveled by car has to travel to catch up to the train is 240 miles.

Given that,

A train travels due north at 40 mph.

A car travels toward the train from the same starting point leaving 2 hours later than the train.

If the car is traveling at 60 mph.

We have to determine,

How many miles will the car have to travel to catch up to the train?

According to the question,

The distance will the car have to travel to catch up to the train is determined by using the following formula;

[tex]\rm Speed = \dfrac{Distance}{Time}[/tex]

Let the distance will the car have to travel to catch up to the train be x.

Then,

The distance traveled by car in times t is,

[tex]\rm Speed = \dfrac{Distance}{Time}\\\\Distance = Speed \times Time\\\\ x = 40 \times t\\\\x = 60t[/tex]

And car travels toward the train from the same starting point leaving 2 hours later than the train.

Then,

The distance traveled by train in time (t+2) is,

[tex]\rm Speed = \dfrac{Distance}{Time}\\\\Distance = Speed \times Time\\\\ x = 40 \times( t+2)\\\\x = 40(t+2)[/tex]

Substitute the value of x in equation 2 from equation 1,

[tex]\rm x = 40(t+2)\\\\60t=40t+80\\\\60t-40t=80\\\\20t=80\\\\t = \dfrac{80}{20}\\\\t = 4[/tex]

Therefore,

The distance traveled by car has to travel to catch up to the train is,

[tex]\rm x = 60t\\\\x = 60\times 4\\\\x = 240 \ miles[/tex]

Hence, The distance traveled by car has to travel to catch up to the train is 240 miles.

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Answer:

yes it is

Step-by-step explanation:

Answer: 53.

Step-by-step explanation:

The given expression : [tex]8-\dfrac{m}{n}+p^2[/tex]

To find : The value of the above expression for the values m=8, n=2 and p=7.

For that , we just substitute the gives values m=8, n=2 and p=7 in the above expression by using Substitution property  , we get

[tex]8-\dfrac{8}{2}+(7)^2[/tex]  

Simplify,

[tex]=8-4+49[/tex]  

[tex]=4+49=53[/tex]    

Hence, the correct value of the given expression when m=8 n=2 p=7 is 53.

Final answer:

To calculate the remaining travel time, subtract the distance already driven from the total distance and divide the result by the speed. It will take 3 hours to reach the destination.

Explanation:

The question involves calculating the remaining travel time to reach a destination, given the current distance traveled, the total distance, and the traveling speed. We're provided with the equation 55h + 275 = 440, where h represents the number of hours needed to reach the destination.

Here's how we solve the equation step-by-step:

Subtract 275 from both sides of the equation to isolate the term with h. This gives us 55h = 440 - 275.

Subtract 275 from 440 to get the remaining distance, which is 165 miles.

Divide the remaining distance by the speed to find the time in hours. So we have h = 165/55.

Calculate the division to find h = 3 hours.

Therefore, it will take 3 hours to reach the destination after having already driven 275 miles at a speed of 55 miles per hour.

4 seconds

Let they meet after time 't' seconds

Let distance covered by puppy in 't' seconds be 'x' meters

So distance covered by kitten in 't' seconds is (180 - x ) meters

Time = Distance / Speed

so,

t = x / 25   for puppy

t = (180-x) / 20     for kitten

On solving,

x = 100 meters

t = 4 seconds

Answer:  Sum of the geometric series will be [tex]\frac{763}{972}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\frac{1}{4}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}+\frac{16}{243}[/tex]

Here,

[tex]a=\frac{2}{9}\\\\r=\frac{a_2}{a_1}\\\\r=\frac{\frac{4}{27}}{\frac{2}{9}}=\frac{4}{27}\times \frac{9}{2}=\frac{2}{3}\\\\n=4[/tex]

As we know the formula for "Sum of n terms in geometric series ":

[tex]S_n=\frac{a(1-r^n)}{1-r}\\S_n=\frac{\frac{2}{9}(1-\frac{2}{3}^4)}{1-\frac{2}{3}}\\S_n=\frac{130}{243}[/tex]

So, Complete sum  will be

[tex]\frac{130}{243}+\frac{1}{4}=\frac{520+243}{972}=\frac{763}{972}[/tex]

Hence, Sum of the geometric series will be [tex]\frac{763}{972}[/tex]

To find the total length of four 3/8-inch pieces, you multiply the length of one piece (3/8 inch) by the number of pieces (4), which equals 1.5 inches.

To calculate the total length of all the 3/8-inch pieces, we multiply the length of one piece by the number of pieces, which is 4. The mathematical expression for this is:

Total length = length of one piece x number of pieces

Total length = 3/8 x 4

To perform the multiplication, multiply the numerators and then the denominators:

Total length = (3 x 4)/(8 x1)

Total length = 12/8 inches

Now, simplify the fraction by dividing the numerator and the denominator by the greatest common divisor, which is 4:

Total length = (12/4)/(8/4)

Total length = 3/2 inches

And since 3/2 inches is equal to 1.5 inches, the total length of all four pieces is 1.5 inches.

You can use the fact that the range of tangent function is whole set of real numbers.
The solutions to the given equation are
[tex]x = \pm \dfrac{\pi}{3 }+ n\pi ; \: n \in \mathbb Z\\[/tex]

[tex]sin^2(\theta) + cos^2(\theta) = 1\\\\1 + tan^2(\theta) = sec^2(\theta)\\\\1 + cot^2(\theta) = csc^2(\theta)[/tex]

It is a fact that tangent ratio has range as all real numbers. We can use this fact along with the second Pythagorean identity to get to the solution of the given equation.

The given equation is [tex]2sec^2x-tan^4x=-1[/tex]

Using the second Pythagorean identity, we get the equation as

[tex]2\sec^2x-\tan^4x=-1\\\\2(1 + \tan^2x) - (\tan^2x)^2= -1\\\\(\tan^2x)^2 -2\tan^2x -3 = 0[/tex]

Assuming [tex]y = tan^2x[/tex], then we get [tex]y \geq 0[/tex]

The equation becomes

[tex](\tan^2x)^2 -2\tan^2x -3 = 0\\\\y^2 - 2y - 3 = 0\\y-3y + y - 3 = 0\\y(y - 3) + 1(y-3) = 0\\(y+1)(y-3) = 0\\y = -1, y = 3[/tex]

As we know that y is non-negative, so only valid solution is y = 3

Thus,

[tex]y = tan^2(x) = 3\\tan(x) = \pm \sqrt{3}\\x = \tan^{-1}(\pm \sqrt{3})[/tex]

Thus,

[tex]x = tan^{-1}(\sqrt{3}) = 60^\circ + n\pi ; \: n \in \mathbb Z\\\\x = tan^{-1}(-\sqrt{3}) = -60^\circ + n\pi ; \: n \in \mathbb Z[/tex]

Thus, the solutions to the given equation are:

[tex]x = \pm 60^\circ + n\pi ; \: n \in \mathbb Z\\[/tex]

Converting to radians,

[tex]x = \pm \dfrac{\pi}{3 }+ n\pi ; \: n \in \mathbb Z\\[/tex]

Thus,The solutions to the given equation are
[tex]x = \pm \dfrac{\pi}{3 }+ n\pi ; \: n \in \mathbb Z\\[/tex]

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The exact solutions of the equation 2sec^2x-tan^4x=-1 are x = 2nπ and x = (2n+1)π/2 for integer values of n.

The given equation is 2sec^2x-tan^4x=-1.

Let's simplify the equation:

2(1/cos^2x)-(tan^2x)^2 = -1

2/cos^2x - tan^4x = -1

Now, substituting sec^2x = 1/cos^2x and tan^2x = (sinx/cosx)^2, we get:

2(1/cos^2x)-((sinx/cosx)^2)^2 = -1

2/cos^2x - sin^4x/cos^4x = -1

Now, let's substitute sin^2x = 1 - cos^2x:

2/cos^2x - (1-cos^2x)^2/cos^4x = -1

Now, solving for cos^2x:

2/cos^2x - (1-2cos^2x+cos^4x) = -1

2 - 2cos^2x + cos^4x - cos^2x = -cos^2x

cos^4x - 3cos^2x + 2 = 0

Now, we can solve for cos^2x by factoring the quadratic equation:

(cos^2x - 2)(cos^2x - 1) = 0

cos^2x = 2 or cos^2x = 1

Since the range of cos^2x is [0,1], we can discard the solution cos^2x = 2.

Therefore, cos^2x = 1.

Which means, cosx = ±1.

Since the required range is [0,2π], we can take two solutions:

cosx = 1, implies x = 2nπ, where n is an integer.

cosx = -1, implies x = (2n+1)π/2, where n is an integer.

Hence, the exact solutions of the equation 2sec^2x-tan^4x=-1 are x = 2nπ and x = (2n+1)π/2 for integer values of n.

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We have been given that a ship is sailing through the water in the English Channel with a velocity of 22 knots along a bearing of 157°.

Further we have been given that current has a velocity of 5 knots along a bearing of 213°.

Therefore, angle between the direction of ship and direction of current will be

[tex]\theta = 213 - 157 = 56^{0}[/tex]

We can find the magnitude of resultant by using parallelogram law of vectors.

[tex]R=\sqrt{P^{2}+Q^{2}+2PQcos(\theta)}[/tex]

Upon substituting [tex]P=22, Q = 5 \text{ and }\theta = 56[/tex] in this formula, we get

[tex]R=\sqrt{22^{2}+5^{2}+2\cdot 22\cdot 5cos(56)}\\ R=\sqrt{484+25+220\cdot0.55919}\\ R=\sqrt{632.0224}\\ R=25.14 \text{ knots}[/tex]

Therefore, resultant velocity of the ship is 25.14 knots.

We find the angle of resultant from P, that direction of ship using the formula

[tex]\alpha = arctan(\frac{Qsin(\theta)}{P+Qcos(\theta)})[/tex]

Upon substituting the values, we get

[tex]\alpha = arctan(\frac{5sin(56)}{22+5cos(56)})\\ \alpha = arctan(\frac{4.14518}{24.79596})\\ \alpha = arctan(0.16717)\\ \alpha = 9.49^{0}[/tex]

Therefore, bearing of the resultant is [tex]157+9.49 = 166.49^{0}[/tex]

Hence, option (A) is the correct choice!

Answer:

First graph

Step-by-step explanation:

Given function,

[tex]g(x)=\sqrt{x-1}+1[/tex]

∵ The point from which the graph of the function passes through will be satisfy the function,

In the first graph,

Graph is passing through (1, 1), (5, 3) and (10, 4),

[tex]1=\sqrt{1-1}+1[/tex]

[tex]3=\sqrt{5-1}+1[/tex]

[tex]4=\sqrt{10-1}+1[/tex]

So, all points (1, 1), (5, 3) and (10, 4) satisfy the function,

In the second graph,

Graph is passing through (-1, -1), (3,1) and (8,2),

Since,

[tex]-1\neq \sqrt{-1-1}+1[/tex]

So, all points of graph 2 do not satisfy the function,

In the third graph,

Graph is passing through (1, -1), (5, 1) and (10, 2),

[tex]-1\neq \sqrt{1-1}+1[/tex]

So, all points of graph 3 do not satisfy the function,

In the fourth graph,

Graph is passing through (-1, 1), (3, 3) and (8, 4),

[tex]1\neq \sqrt{-1-1}+1[/tex]

So, all points of graph 4 do not satisfy the function

Applying Pythagorean Identity, the original equation sin² theta +cos theta = 2 can be altered by substituting sin² theta with 1 - cos² theta. This forms a quadratic equation: cos² theta - cos theta - 1 = 0, which can be solved using the quadratic formula to find the possible values of cos theta.

To solve the equation sin² theta +cos theta = 2 by using the Pythagorean identity sin² theta +cos² theta =1, we can substitute sin² theta in the original equation with 1 - cos² theta.

Step-by-step solution:

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Answer:

Option A is correct

Value of A is, 189

Step-by-step explanation:

Joint variation states:

A varies jointly as b and c.

⇒[tex]A \propto b[/tex] and [tex]A \propto c[/tex]

⇒[tex]A \propto bc[/tex]

then the equation is in the form of:

[tex]A = k \cdot bc[/tex]

where, k is the constant of variation.

As per the statement:

A varies jointly as b and c.

⇒[tex]A = k \cdot bc[/tex]           ....[1]

To find A .

Substitute the given values b = 7 , c = 9 and k = 3 in [1]

then;

[tex]A = 3 \cdot 7 \cdot 9 = 21 \cdot 9 = 189[/tex]

Therefore, the value of A is, 189

After 3 half-lives, approximately 14 grams of the radioactive isotope will be left.

We have,

To calculate the remaining mass of a radioactive isotope after a certain number of half-lives, we can use the formula:

Remaining Mass = Initial Mass * (1/2)^(Number of Half-Lives)

Given:

Initial Mass = 115 grams

Number of Half-Lives = 3

Substituting the values into the formula, we get:

Remaining Mass = 115 * (1/2)^3

Calculating this expression:

Remaining Mass = 115 * (1/2)³

Remaining Mass = 115 * (1/8)

Remaining Mass = 14.375

Rounding to the nearest gram, the remaining mass is approximately 14 grams.

Therefore,

After 3 half-lives, approximately 14 grams of the radioactive isotope will be left.

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The probability that he will win this week is 0.0001%.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.

The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.

The degree to which something is likely to happen is basically what probability means.

Given:

Ten ping-pong balls numbered 0 to 9 are placed in each of six containers.

It will not effect the outcome if Juan didn't win last week.

So, the probability that he will win this week

= [tex](1/10)^6[/tex]

= 1/1,000,000

= 0.000001

= 0.0001%

Hence, the probability that he will win this week is 0.0001%.

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With the deposit of 3000, the balance after 2 years would be $3307.50

How do we find the compound interest for the deposit?

The formula is: A = P(1+r)ⁿ

A is the amount of money accumulated after n years, including interest.

P = $3000

r = 5% = 0.05 (as a decimal)

n = 2 years

A = 3000 × (1+0.05)²

A = 3000 × (1.05)²

A = 3000 × 1.1025

A = 3307.50

Therefore, when a deposit of $3000 is made, the balance after 2 years would be $3307.50

Final answer:

The probability of independent events A, B, and C all occurring together is found by multiplying their individual probabilities: P(A AND B AND C) = P(A) × P(B) × P(C) = 0.090528.

Explanation:

The question involves calculating the probability of all three independent events happening simultaneously. To find the combined probability of independent events A, B, and C occurring at the same time, we use the multiplication rule of probability.

This rule states that if the events are independent, the probability of all events occurring is the product of their individual probabilities.

Here are the probabilities for each event:

Probability of event A (P(A)) = 0.43

Probability of event B (P(B)) = 0.32

Probability of event C (P(C)) = 0.66

The probability of events A, B, and C all occurring together is:

P(A AND B AND C) = P(A) × P(B) × P(C) = (0.43) × (0.32) × (0.66) = 0.090528