In a standard deck of cards there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and the clubs are black and the hearts and the diamonds are red.

If two cards are chosen at random from a deck, one at a time, and replaced after each pick, what is the probability that a black card is chosen first and a heart is chosen second?
-1/8
-1/2
-2/3
-3/4

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.

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The required height of the trapezoid is 11.5 inches.

Simplification involves applying rules of arithmetic and algebra to remove unnecessary terms, factors, or operations from an expression. The goal is to obtain an expression that is easier to work with, manipulate, or solve.

Here,
The formula for the area of a trapezoid is:

A = (b₁ + b₂) * h / 2

where A is the area, b1 and b2 are the lengths of the two bases, and h is the height.

We are given that the area of the trapezoid is 166.75 in² and the lengths of the two bases are 21 in and 8 in. So we can plug these values into the formula and solve for h:

166.75 = (21 + 8) * h / 2

Simplifying the expression on the right side:

166.75 = 29 * h / 2

Multiplying both sides by 2:

333.5 = 29 * h

Dividing both sides by 29:

h = 11.5

So the height of the trapezoid is 11.5 inches.

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Final answer:

To find the height of a trapezoid given its area (166.75 sq in) and bases (21 in and 8 in), you use the formula-

[tex]A = \frac{1}{2} h(b_1 + b_2)[/tex]

By rearranging to solve for h, you find that the height is 11.5 inches.

Explanation:

The question asks to find the height of a trapezoid given its area and bases. The area of a trapezoid is calculated using the formula A = [tex]\(\frac{1}{2}\)h(b_1 + b_2)[/tex], where A represents the area, h the height, and [tex]b_1[/tex] and [tex]b_2[/tex] the lengths of the two bases. In this case, the area is 166.75 square inches, and the bases measure 21 inches and 8 inches.

To find the height, we can rearrange the formula to-

h = [tex]\(\frac{2A}{b1 + b2}\)[/tex]

Plugging in the given values gives us-

h = [tex]\(\frac{2 \times 166.75}{21 + 8}\)[/tex]

which simplifies to-

h = [tex]\(\frac{333.5}{29}\)[/tex]

Therefore, the height of the trapezoid is 11.5 inches.

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.

Answer:

Lateral area=100 square inches

Step-by-step explanation:

It is given that the perimeter of the base of a pentagonal prism is 25 inches, therefore each sides of the pentagon will be of 5 inches.

Also, one of its lateral edge is 4 inches, therefore each lateral face has area which will be equal to=[tex]5{\times}4[/tex]=[tex]20[/tex]square inches.

Since, there are 5 lateral faces of the pentagonal prism, therefore the lateral area of the prism will be: [tex]5{\times}20[/tex]

                                       = [tex]100[/tex]square inches.

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

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.

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:

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

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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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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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

Answer:

15/35 = 3/7 = 0.428571

Explanation:

We are asked to find the probability that a given voter is conservative given they are in State B.

Going to the State B column, we see that there are 15 conservatives.  This is out of a total of 35 voters in state B; this makes the probability 15/35.

Probability helps us to know the chances of an event occurring. The probability that a voter from State B is supporting Conservative is 0.4285.

Probability helps us to know the chances of an event occurring.

[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]

We're supposed to figure out how likely it is that a specific voter in State B is a conservative.

There are 15 conservatives in State B, according to the column. This is based on a total of 35 voters in state B, resulting in a 15/35 probability. Therefore,

P(conservative | State B) = 15/35 = 3/7 = 0.4285

Hence, the probability that a voter from State B is supporting Conservative is 0.4285.

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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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Answer:

-6,-6

Step-by-step explanation:

(-9,2) + (-8,3) = (-6,-6)

correct on the test :)