Eric practice piano and guitar for 8 hrs. He practiced piano for 1 fourth of 8 hrs. How many hours did Eric practice the piano this week?

Answer:

  d. 11.3 cm

Step-by-step explanation:

The radius of the circle is the length CP, which can be found using the Pythagorean theorem. Since CQ ⊥ PR, you know that Q is the midpoint of PR and PQ = 8 cm.

Then the Pythagorean theorem tells you ...

  CP² = CQ² +PQ² = (8 cm)² + (8 cm)² = 128 cm²

  CP = √128 cm = 8√2 cm

  CP = 11.3 cm

Answer:

[tex]\large\boxed{d.\ 11.3\ cm}[/tex]

Step-by-step explanation:

Look at the picture.

CQ is a perpendicular bisector of PR. Therefore QR = QP.

[tex]PR=16\ cm\to QP = 16 cm : 2 = 8\ cm[/tex]

The segment CP is a radius of the given circle.

We have the right triangle CQP. Use the Pythagorean theorem:

[tex]CQ^2+QP^2=CP^2[/tex]

Substitute CQ = 8 cm and QP = 8 cm.

[tex]CP^2=8^2+8^2\\\\CP^2=64+64\\\\CP^2=128\to CP=\sqrt{128}\\\\CP\approx11.3\ cm[/tex]

Answer:

56 divided by 8 would equal p.

Step-by-step explanation:

If you want to find how many packs there are, it would be 56 divided by 8 would equal p.

Brainiest Please :D

Answer:

The equation is [tex]p*8=56[/tex]

Step-by-step explanation:

The neighbors buy a number of packs (p) of hot dogs and each one has 8 hot dogs. To calculate the number of hot dogs resulting we have to multiply the packs (p) by the number of hot dogs in each pack (8), this is:

[tex]p*8=56[/tex]

Then to calculate p we have to clear it:

[tex]p=\frac{56}{8} =7[/tex]

The number of packs bought were 7.

The length of EF is 3.8. Option A

To determine the value of the side, we have to make use of the sine rule, this rule is represented as;

sin A/a = sin B/b

Such that;

Now, substitute the values, we have;

sin 75/EF = sin 50/3

cross multiply the values, we get;

EF = 3sin 75/sin 50

find the values

EF = 3(0.96)/0.766

EF = 2.88/0.766

Divide the values

EF = 3. 8

Answer:

w√26

Step-by-step explanation:

A rectangle is a four sided shape with 4 perpendicular angles. It has two pairs of parallel sides which are equal in distance: width and length. The width here is w and the length is 5w or 5 times the width. A diagonal can be drawn between opposite corners that splits the triangle into two equal right triangles. The distance of this diagonal is found using the Pythagorean Theorem a² + b² = c². In the rectangle a = w and b = 5w. Substitute these values and simplify using a square root operation.

w² + (5w)² = c²

w² + 25w² = c²

26w² = c²

√26w² = c

w√26 = c

Answer: The Slope-Intercept Form equation is y=mx+b

i think the answer is y=3x+2 but i hope this helps

The equation of the line in Slope-intercept form is y = -3x + 2.

The correct choice is option C. y=-3x+2.

Let's, find the equation of the line in slope-intercept form, we need to determine the slope (m) and the y-intercept (b). The slope represents the steepness of the line, and the y-intercept is the point where the line crosses the y-axis.

From the graph, we can see that the line passes through the points (-1, 5) and (1, -1). Using the slope formula, we can calculate the slope as follows:

m = (y2 - y1) / (x2 - x1)

m = (-1 - 5) / (1 - (-1))

m = -6 / 2

m = -3

Now that we know the slope, we can plug it into the slope-intercept form of the equation and solve for b:

y = mx + b

y = -3x + b

We can find the value of b by substituting the coordinates of one of the points on the line into the equation. For example, we can use the point (-1, 5):

5 = -3(-1) + b

5 = 3 + b

b = 5 - 3

b = 2

Therefore, the equation of the line in slope-intercept form is y = -3x + 2.

option C. is correct choice y=-3x+2.

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now, let's recall that a geometric sequence is one that uses some "r" common ratio to get the next term, by simply multiplying the current term by it.

[tex]\bf \begin{array}{|cl|ll} \cline{1-2} term&value\\ \cline{1-2} a_1&\underline{\qquad }\\&\\ a_2&6\\&\\ a_3&\underline{6(r)}\\&\\ a_4&6(r)(r)\\&\\ &54\\ \cline{1-2} \end{array}\qquad \implies \begin{array}{llll} 54=6r^2\implies \cfrac{54}{6}=r^2\implies 9=r^2\\\\ \sqrt{9}=r\implies 3=r \end{array} \\\\[-0.35em] ~\dotfill\\\\ a_1=6\div 3\implies a_1=2~\hfill a_3=6(3)\implies a_3=18[/tex]

and of course, the next term or a₅ = 54(3) --> a₅ = 162.

The first five terms are 2, 6, 18, 54, 162

Given,

The second term is 6.

The fourth term is 54.

We need to find the first five terms of a geometric sequence.

A sequence where each term after the first is found by multiplying the previous one with a common ratio.

The sequence is given by:

a, ar, ar^2, ar^3, ar^4, ar^5,...

The nth term is given by:

a_n = ar^(n-1)

We have,

Second term = 6

a_2 = ar^(2-1)

6 = ar^1

6 = ar

a = 6/r _____(1)

Fourth term = 54

a_4 = ar^(4-1)

54 = ar^3

a = 54/r^3 ______(2)

From (1) and (2)

6/r = 54/r^3

r^3/r = 54/6

r^2 = 9

r = 3

Putting in (1)

a = 6/r

a = 6/3

a = 2

We have,

a = 2 and r = 3

Find the first five terms of a geometric sequence.

It is given by:

a, ar, ar^2, ar^3, ar^4

2, 2x3, 2x9, 2x27, 2x81

2, 6, 18, 54, 162

The first five terms are 2, 6, 18, 54, 162

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Answer:

13

Step-by-step explanation:

Answer:

Their average rate of speed is 53 miles per hour.

Step-by-step explanation:

Given : A family on a vacation drives 123 miles in 2 hours then gets stuck in traffic and goes 4 miles in the next 15 minutes. The remaining 191 miles of the trip take [tex]3\frac{3}{4}[/tex] hours.

To find : What was their average rate of speed to the nearest tenth of a mile per hour ?

Solution :

We know, [tex]\text{Speed}=\frac{\text{Distance}}{\text{Time}}[/tex]

Total distance traveled by family on vacation is

D= 123 miles + 4 miles + 191 miles = 318 miles

Total time taken by family on vacation is

T= 2 hours + 15 minutes + [tex]3\frac{3}{4}[/tex] hours

T= 2 hours + [tex]\frac{15}{60}[/tex] hours + [tex]3\frac{3}{4}[/tex] hours

T= [tex]2+ \frac{1}{4}+ \frac{15}{4}[/tex] hours

T= [tex]\frac{8+1+15}{4}[/tex] hours

T= [tex]\frac{24}{4}[/tex] hours

T= 6 hours

Substitute the value in the formula,

[tex]\text{Speed}=\frac{318}{6}[/tex]

[tex]\text{Speed}=53[/tex] miles per hour.

Therefore, Their average rate of speed is 53 miles per hour.

Answer:

Step-by-step explanation:

The diagonals in the parallelogram intersect by dividing in half.

Therefore we have the equation:

2y + 22 = 79 - y                 subtract 22 from both sides

2y = 57 - y              add y to both sides

3y = 57              divide both sides by 3

y = 19

Answer:

A) 0.3336; B) 0.8531

Step-by-step explanation:

For part A,

We use the z-score formula for an individual score:

[tex]z=\frac{X-\mu}{\sigma}[/tex]

Our X value is 332, our mean, μ, is 335, and our standard deviation, σ, is 7:

z = (332-335)/7 = -3/7 ≈ -0.43

Using a z table, we see that the area under the curve less than this (the probability that X is less than this value) is 0.3336.

For part B,

We use the z-score formula for the mean of a sample:

[tex]z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}[/tex]

Our X-bar value is 332, our mean, μ, is 335, our standard deviation, σ, is 7, and our sample size, n, is 6:

z = (332-335)/(7÷√6) = 3/2.8577 ≈ 1.05

Using a z table, we see that the are under the curve to the left of this, or the probability less than this, is 0.8531.

Using the Z-score formula, the probability of a single beer bottle having less than 332 ml is 33.36%, and the probability of a 6-pack having a mean amount less than 332 ml is 14.69%.

To solve this problem, we can use the Z-score. The Z-score is the number of standard deviations a particular value is from the mean in a normal distribution. The formula for the Z-score is (X-µ)/σ, where X represents the value of interest, µ represents the population mean, and σ represents the standard deviation.

So, let's calculate the Z-score:
a) We use the formula Z = (X-µ)/σ = (332-335)/7 = -0.43 (rounded to 2 decimal places). To find the probability that a bottle of beer contains less than 332 ml, we refer to the standard Z-table, which gives us approximately 0.3336. Therefore, there is a 33.36% chance a randomly selected beer bottle contains less than 332 ml of beer.

b) For a 6-pack, the standard deviation decreases because it is now σ/√n (with n being the size of the sample, in this case, 6). The new standard deviation is 7/√6 = 2.86 ml (rounded to 2 decimals). Using the same Z-score formula, Z= (332-335)/2.86= -1.05, and referring to the Z-table, the probability is approximately 0.1469. This means there's about a 14.69% chance that a randomly selected 6-pack will have a mean amount of less than 332 ml.

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Answer:

It took 2 hours by plane

Step-by-step explanation:

let bus is represented by x and plane is represented by y

the total distance of bus and plane will be represented by equation:

40 x + 600 y = 1320           (1)

the total time taken by bus and plane will be represented by:

x + y = 5                              (2)

solving equations (1) and (2) simultaneously:

Multiply by 40 on both sides in eq (2)  and then subtract (1) and (2)

40 x + 600 y = 1320

40 x + 40 y = 200

_______________

          560 y = 1120

                y   = 1120/ 56

                y= 2

since plane is represented by y so, the time they spent travelling by plane is 2 hours.

Final answer:

By setting up and solving a system of equations based on the total travel time and distance, we find that the time spent travelling by plane is 2 hours.

Explanation:

To determine the time spent travelling by plane, we start by setting up two equations based on the given information.

Let x represent the time spent on the bus and y represent the time spent on the plane. The total travel time is given as 5 hours, and the total distance travelled is 1320 km.

Therefore, we have two equations:

We can solve this system of equations by first expressing x in terms of y from the first equation:

x = 5 - y

Substituting x in the second equation:

40(5 - y) + 600y = 1320

Solving for y, we find that the time spent travelling by plane is 2 hours.

Answer:

Step-by-step explanation:

can you explain a little bit more ?

Answer:

It will take 5 years for the population to reach 1,944. Option D is correct.

Step-by-step explanation:

Initial population = I = 8

Final population = F = 1944

number of years = x

As given, the population is tripled every year that can be calculated using:

F = I*3^x

By putting values in this equation, we get

1944 = 8*3^x

1944/8 = 3^x

243 = 3^x

3^x = 243

3^x = 3^5

x = 5 bases are same so powers can be simplified.

Therefore, it will take 5 years for the population to reach 1,944. Option D is correct.

Answer:

8 • 3^y = R

Step-by-step explanation:

...................

Answer:

Step-by-step explanation:

The question is incomplete. p(x 16) is actually [tex]P(X\geq 16)[/tex] ; p(x 15) is actually [tex]P(X\leq 15)[/tex] and e(x) is [tex]E(X)[/tex]

Wherever a random variable X can be modeled as a binomial random variable we write :

X ~ Bi (n,p)

Where ''n'' is the number of Bernoulli experiments taking place (whose variable is called binomial random variable).

And where ''p'' is the success probability.

In a Bernoulli experiment we define which event will be a ''success''

In order to calculate the probabilities for the variable X we can use the following equation :

[tex]P(X=x)=f(x)=(nCx).(p^{x}).(1-p)^{n-x}[/tex]

Where ''[tex]P(X=x)[/tex]'' is the probability of the variable X to assume the value x.

Where ''[tex]nCx[/tex]'' is the combinatorial number define as :

[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]

In our question

X ~ Bi (20,0.70)

Now let's calculate the probabilities :

[tex]f(12)=P(X=12)=(20C12).(0.70)^{12}.(1-0.70)^{20-12}=0.1144[/tex]

[tex]f(16)=P(X=16)=(20C16).(0.70)^{16}.(1-0.70)^{20-16}=0.1304[/tex] (I)

[tex]P(X\geq 16)[/tex] ⇒

[tex]P(X\geq 16)=P(X=16)+P(X=17)+P(X=18)+P(X=19)+P(X=20)[/tex] (II)

[tex]P(X=17)=(20C17).(0.70)^{17}.(1-0.70)^{20-17}=0.0716[/tex] (III)

[tex]P(X=18)=(20C18).(0.70)^{18}.(1-0.70)^{20-18}=0.0278[/tex] (IV)

[tex]P(X=19)=(20C19).(0.70)^{19}.(1-0.70)^{20-19}=0.0068[/tex] (V)

[tex]P(X=20)=(20C20).(0.70)^{20}.(1-0.70)^{20-20}=0.0008[/tex] (VI)

Using (I), (III), (IV), (V) and (VI) in (II) :

[tex]P(X\geq 16)=0.1304+0.0716+0.0278+0.0068+0.0008=0.2374[/tex]

Now :  

[tex]P(X\leq 15)[/tex]

[tex]P(X\leq 15)=1-P(X\geq 16)[/tex]

[tex]P(X\leq 15)=1-0.2374=0.7626[/tex]

Finally,

[tex]E(X)=[/tex] μ (X)

[tex]E(X)[/tex] is the mean of the variable X

In this case, X is a binomial random variable and its mean can be calculated as

[tex]E(X)=(n).(p)[/tex]

In the question :

[tex]E(X)=(20).(0.70)=14[/tex]

The binomial experiment with n = 20 and p = 0.70 indicates that the probabilities are;

f(12) ≈ 0.1144

f(16) ≈ 0.1304

P(X ≥ 16) ≈ 0.2375

P(X ≤ 15) ≈ 0.7265

E(X) = 14

A binomial experiment is a statistical experiment that consists of a specified number of independent trials, in which each the trials has only two possible outcomes. The probability of success is the same for all trials and the trials are independent, such that the outcome of one trial does not affect the outcome of the other trials.

In a binomial experiment with n = 20 and p = 0.7, the probability of exactly k successes in n independent trials can be found from the following probability mass function.

f(k) = [tex]_nC_k[/tex] × [tex]p^k[/tex] × [tex](1 - p)^{(n-k)}[/tex]

Where [tex]_nC_k[/tex] is the binomial coefficient, which can be calculated an [tex]_nC_k[/tex] = n!/(k!·(n - k)!)

Using the formula the probabilities can be calculated as follows;

f(12) = ₂₀C₁₂ × 0.70¹² × 0.3⁸ = 125970 × 0.70¹² × 0.3⁸ ≈ 0.1144

f(16) = ₂₀C₁₆ × 0.70¹⁶ × 0.3⁴ = 4845 × 0.70¹⁶ × 0.3⁴ ≈ 0.1304

P(X ≥ 16) = f(16) + f(17) + f(18) + f(19) + f(20) ≈ 0.2375

P(X ≤ 15) = 1 - P(X ≥ 16) ≈ 1 - 0.2375 = 0.7625

The expected value of the binomial random variable X is; E(X) = n·p, where n is the number of trials and p is the probability of success on a single trial.

E(X) = n·p = 20 × 0.7 = 14

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An example of dependent events is drawing a red marble from the same jar without replacing the first marble, which changes the probabilities for the second draw. This is known as sampling without replacement, which contrasts with independent events where the item is replaced and the probabilities remain unchanged.

An example of dependent events is drawing a red marble out of the same jar, without replacing the first marble. This creates a dependency because removing the first marble affects the outcome chances for the subsequent draw. If the jar starts with four blue and three white marbles and you draw one blue marble and do not replace it, the jar then contains three blue and three white marbles. Therefore, the probability of drawing another blue marble has changed from the original draw.

Dependent events are associated with sampling without replacement, which means once an item is drawn, it is not put back into the population from which it was taken, altering the probabilities of subsequent draws. In contrast, independent events, such as drawing and replacing a marble, do not affect the subsequent probabilities, since the composition of the jar remains unchanged with each draw.

The line parallel to the equation 2x + 3y = 12 can be expressed in the form y = -2/3x + c, where c is any arbitrary constant. This is because parallel lines share the same slope.

In mathematics, parallel lines have the same slope. To find an equation that is parallel to an existing line, we need to find the slope of the existing line and use that same slope for the parallel line. We know that your line equation is 2x + 3y =12. Let's rewrite this in the y = a + bx form, where a is the y-intercept and b is the slope.

So, subtract 2x from both sides to get the equation in the form of 3y = -2x +12. Now, divide the entire equation by 3 to isolate y. You get y = -2/3x + 4. Here, the slope (b) is -2/3. Any line parallel to this one must also have a slope of -2/3.

Thus, the parallel line is y = -2/3x + c, where c is any arbitrary constant.

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To find a line parallel to 2x + 3y = 12, you must use the same slope, which is -2/3. Any line of the form y = (-2/3)x + b, where b is any number, will be parallel to the original line.

To find an equation for a line parallel to the given line 2x + 3y = 12, we need to find a line with the same slope. First, we rearrange the given equation into slope-intercept form (y = mx + b), where m represents the slope, and b represents the y-intercept.

The original equation can be rewritten as:

3y = -2x + 12

y = (-2/3)x + 4

The slope of this line is -2/3. Any line parallel to this line will have the same slope, -2/3. The general form of the equation for any parallel line will be:

y = (-2/3)x + b

where b can be any real number.

Here are a couple of examples of lines that are parallel to the original:

Both have the same slope (-2/3) as the original line but different y-intercepts.

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Answer: ❤️Hello!❤️He asks every tenth customer who checks into the hotel. Hope this helps! ↪️ Autumn ↩️

Answer: 321.4167 or about 322 boxes

Step-by-step explanation:

➷ Find the total number of cards:

24 + 18 + 18 = 60

Find the total number of cards that aren't green:

24 + 18 = 42

The probability would be 42/60

This could be simplified to 7/10

➶ Hope This Helps You!

➶ Good Luck (:

➶ Have A Great Day ^-^

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Answer:

710

Step-by-step explanation:

7/10

Answer:

This would probely be 266.

Step-by-step explanation:

There are three kids so I would divide them by three. Or multiply 266 by three. Of course there would still be 2$ left.

Answer:

Share of Eli, Freda and Geoff will be $180, $260, $360 respectively.

Step-by-step explanation:

Eli, Freda and Geoff were given $800 to share in the ratio of their age.

Their ages were 9 years, 13 years and 18 years respectively.

Ratio of their ages will be 9 : 13 : 18

Now share of Eli will be = [tex]\frac{9}{9+13+18}\times 800[/tex]

= [tex]\frac{9\times 800}{40}[/tex]

= $180

Share of Freda will be = [tex]\frac{13}{9+13+18}\times 800[/tex]

                                     = $260

Share of Geoff will be = [tex]\frac{18}{9+13+18}\times 800[/tex]

                                     = $360

Therefore, share of Eli, Freda and Geoff will be $180, $260, $360 respectively.

Answer:

[tex]1\frac{2}{3}\ kg[/tex]

Step-by-step explanation:

we know that

To find out how much the guests ate, multiply the total amount of kg of mashed potatoes by the 4/12 fraction

so

[tex]5(\frac{4}{12})=\frac{20}{12}\ kg[/tex]

convert to mixed number

[tex]\frac{20}{12}=\frac{12}{12}+\frac{8}{12}=1\frac{8}{12}\ kg[/tex]

simplify

[tex]1\frac{8}{12}=1\frac{2}{3}\ kg[/tex]

The guests consumed 1.67 kilograms of the 5 kilograms of mashed potatoes that the chef had made.

To solve this problem, we need to multiply the total amount of mashed potatoes made by the fraction that the guests consumed.

Given that the chef cooked 5 kilograms of mashed potatoes and the guests ate 4/12 (which simplifies down to 1/3) of this amount, we multiply these two together.

So, the calculation would be 5 × 1/3 = 1.67 kilograms.

Therefore, the guests ate 1.67 kilograms of mashed potatoes.

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Answer:

Part a) [tex]V=1,000\ ft^{3}[/tex]

Part b) [tex]h=5\ ft[/tex]

Part c) [tex]=718\ gal[/tex]

Step-by-step explanation:

Part a) What is the volume of your friend's swimming pool?

we know that

The volume of a rectangular prism is equal to

[tex]V=LWH[/tex]

substitute the values

[tex]V=(25)(8)(5)=1,000\ ft^{3}[/tex]

Part b) Your swimming pool is in the shape of a cylinder with a diameter of 16 feet and has the same volume as your friend's pool. What is the height of your pool?

we know that

The volume of a cylinder is equal to

[tex]V=\pi r^{2} h[/tex]

we have

[tex]V=1,000\ ft^{3}[/tex]

[tex]r=16/2=8\ ft[/tex] ----> the radius is half the diameter

substitute and solve for h

[tex]1,000=\pi (8)^{2} h[/tex]

[tex]h=1,000/(64 \pi)[/tex]

[tex]h=1,000/(64*3.14)=5\ ft[/tex]

Part c) While you were on vacation, 6 inches of water evaporated from your pool. About how many gallons of water evaporated from your pool?

remember that

[tex]1\ ft=12\ in[/tex]

so

[tex]6\ in=6/12=0.5\ ft[/tex]

Find the volume of water  for [tex]h=5-0.5=4.5\ ft[/tex]

[tex]V=(3.14)(8)^{2}(4.5)=904.32\ ft^{3}[/tex]

so

the volume of water evaporated is equal to

[tex]1,000\ ft^{3}-904.32\ ft^{3}=95.68\ ft^{3}[/tex]

convert to gallons

remember that

[tex]1\ ft^{3}=7.5\ gal[/tex]

so

[tex]95.68\ ft^{3}=95.68*7.5=718\ gal[/tex]

a. The volume of the rectangular prism is 1000 cubic feet. b. The height of the cylindrical pool is 4 feet. c. About 750 gallons of water evaporated from the pool.

a. The volume of the rectangular prism can be found by multiplying the length, width, and height. So, the volume is 25 * 8 * 5 = 1000 cubic feet.

b. Since the two pools have the same volume, we can use the formula for the volume of a cylinder (V = πr²h) to find the height of the cylinder. We know the diameter is 16 feet, so the radius is half of that, which is 8 feet. Plugging it into the formula, 1000 = 3.142 * 8² * h. Solving for h, we find h = 4 feet.

c. To find the gallons of water evaporated, we need to convert the units. 6 inches is 0.5 feet. The volume evaporated is 25 * 8 * 0.5 = 100 cubic feet. Since 1 cubic foot is equal to 7.5 gallons, the amount evaporated is 100 * 7.5 = 750 gallons.

Answer:

  It depends

Step-by-step explanation:

The shape will have a number of sides equal to the number of faces it intersects (counting the base). Depending on the shape of the hexagonal base, the symmetry of the peak, and where the slicing is done, the vertical slice could have 3, 4, 5, 6, or 7 sides, so could be anything from a triangle to a heptagon.

Answer:

The value of x is 9

Step-by-step explanation:

* It first we must to talk about the rational expression

- The rational expression is defined for all values of x except

  the values make the denominator = zero

Ex: If the rational expression is a/b, then b ≠ 0 to be defined

* In any rational expression we must to avoid any values

 make denominator = zero

* If we want to make rational expression equal to zero then we

  must put the numerator equal to zero

* Now lets look to our problem

∵ The rational expression (x - 9)/(x - 4)(x + 4)

- To make this rational expression equal to zero, we must

  equate the numerator by zero

∴ x - 9 = 0 ⇒ add 9 to both sides

∴ x = 9

* The value of x is 9

* Remember ⇒ x must not equal to 4 or -4 to avoid make

  this rational expression undefined

The distance between the parallel lines 3x - 5y = 1 and 3x - 5y = -3 is 4/√34 units.

To find the distance between two parallel lines with equations 3x - 5y = 1 and 3x - 5y = -3, we need to use the formula for distance between two parallel lines in a coordinate plane. This formula is:

Distance = |C2 - C1| / sqrt(A^2 + B^2)

In this case, your A coefficient is 3, the B coefficient is -5, and the constants (C1 and C2) are -1 and 3 respectively (from the given equations). Substituting these values into the formula gives:

Distance = |3 - (-1)| / sqrt(3^2 + (-5)^2)

Distance = 4 / sqrt(9 + 25)

Distance = 4 / sqrt(34)

So, the distance between parallel lines with equations 3x - 5y = 1 and 3x - 5y = -3 is 4/sqrt(34) units.

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Final answer:

The distance between the parallel lines described by the equations 3x-5y=1 and 3x-5y=-3 is calculated using a standard formula for the distance between parallel lines, resulting in a distance of 4 / √34.

Explanation:

The distance between two parallel lines can be calculated using a formula once the equations of the lines are in standard form. The equations 3x-5y=1 and 3x-5y=-3 are indeed in standard form and clearly show that the lines are parallel because they have the same coefficients for x and y. To find the distance between these parallel lines, one can use the formula involving the absolute difference of the constants in the equations and the square root of the sum of squares of the coefficients of x and y.

The formula for the distance d between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by:

d = |C2 - C1| / √(A² + B²)

Applying this to our equations, we have:

A = 3, B = -5, C1 = 1, and C2 = -3.

Thus,

d = |-3 - 1| / √(3² + (-5)²) = 4 / √(9 + 25) = 4 / √34

Distance between parallel lines, standard form, and formula are essential in solving this problem.

Answer:

[tex]{x}^{7} - 4 {x}^{5} + 2 {x}^{4} + 4 {x}^{3} - 10[/tex]

Step-by-step explanation:

To write a polynomial in descending order means writing the polynomial in decreasing powers of x.

The given polynomial expresion is

[tex] - 4 {x}^{5} + 4 {x}^{3} + {x}^{7} - 10 + 2 {x}^{4} [/tex]

We start from the term with the highest degree and end with term with the least degree.

[tex] {x}^{7} - 4 {x}^{5} + 2 {x}^{4} + 4 {x}^{3} - 10[/tex]

Therefore the given polynomial written in descending order is

[tex]{x}^{7} - 4 {x}^{5} + 2 {x}^{4} + 4 {x}^{3} - 10[/tex]

Answer:

Step-by-step explanation:

see attached

a)

[tex]\sin (\dfrac{7\pi}{4})=-\dfrac{1}{\sqrt{2}}[/tex]

b)

[tex]\cos \dfrac{7\pi}{4}=\dfrac{1}{\sqrt{2}}[/tex]

c)

[tex]\tan \dfrac{7\pi}{4}=-1[/tex]

We are asked to find the value of:

a)

[tex]\sin \dfrac{7\pi}{4}[/tex]

We know that:

[tex]\dfrac{7\pi}{4}=2\pi-\dfrac{\pi}{4}[/tex]

Hence, we have:

[tex]\sin (\dfrac{7\pi}{4})=\sin (2\pi-\dfrac{\pi}{4})\\\\\\\sin (\dfrac{7\pi}{4})=-\sin (\dfrac{\pi}{4})[/tex]

Since,

[tex]\sin (2\pi-\theta)=-\sin \theta[/tex]

Hence, we have:

[tex]\sin (\dfrac{7\pi}{4})=-\dfrac{1}{\sqrt{2}}[/tex]

b)

[tex]\cos \dfrac{7\pi}{4}[/tex]

[tex]\cos (\dfrac{7\pi}{4})=\cos (2\pi-\dfrac{\pi}{4})\\\\\\\cos (\dfrac{7\pi}{4})=\cos (\dfrac{\pi}{4})[/tex]

Since,

[tex]\cos (2\pi-\theta)=\cos \theta[/tex]

Hence, we have:

[tex]\cos \dfrac{7\pi}{4}=\dfrac{1}{\sqrt{2}}[/tex]

c)

[tex]\tan \dfrac{7\pi}{4}[/tex]

[tex]\tan (\dfrac{7\pi}{4})=\tan (2\pi-\dfrac{\pi}{4})\\\\\\\tan (\dfrac{7\pi}{4})=\tan (\dfrac{\pi}{4})[/tex]

Since,

[tex]\tan (2\pi-\theta)=-\tan \theta[/tex]

Hence, we have:

[tex]\tan \dfrac{7\pi}{4}=-1[/tex]

In spherical coordinates, we set

[tex]x=\rho\cos\theta\sin\varphi[/tex]

[tex]y=\rho\sin\theta\sin\varphi[/tex]

[tex]z=\rho\cos\varphi[/tex]

so that the volume element under this transformation becomes

[tex]\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=|\det\mathbf J|\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

The region [tex]E[/tex] is given by the set

[tex]\left\{(\rho,\theta,\varphi)\mid0\le\rho\le1,0\le\theta\le2\pi,0\le\varphi\le\dfrac\pi3\right\}[/tex]

so that the integral is

[tex]\displaystyle\iiint_Ey^2z^2\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi/3}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=1}\rho^6\sin^2\theta\sin^3\varphi\cos^2\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

[tex]\displaystyle=\left(\int_0^{\pi/3}\sin^3\varphi\cos^2\varphi\,\mathrm d\varphi\right)\left(\int_0^{2\pi}\sin^2\theta\,\mathrm d\theta\right)\left(\int_0^1\rho^6\,\mathrm d\rho\right)[/tex]

[tex]=\dfrac{47}{480}\cdot\pi\cdot\dfrac17=\dfrac{47\pi}{3360}[/tex]

The volume of the solid using the triple integral [tex]\mathbf{\iiint_E y^2z^2 dV \ \ is \ \ = \dfrac{47 \pi}{3360}}[/tex]

Given that;

Then the spherical coordinates can be expressed as:

where;

Now, the expression for the solid E in the spherical coordinates can be computed as:

[tex]\mathbf{E = \Big \{ ( \rho , \theta, \phi )\Big| 0 \leq \rho \leq 1, 0 \leq \phi \leq \dfrac{\pi}{3}, 0 \leq \theta \leq 2 \pi \Big \} }[/tex]

and the volume of the solid using the triple integral is calculated as:

[tex]\mathbf{\iiint_E y^2z^2 dV = \iiint _E \ y^2 z^2 \ dx dy dz }[/tex]

[tex]\mathbf{\implies \iiint _E \ y^2 z^2 \ dx dy dz = \int ^{2 \pi}_{0} \int ^{\dfrac{\pi}{3}}_{0} \int ^1_0 \ (\rho sin \phi sin \theta )^2 ( \rho cos \phi )^2 \rho^2 sin \phi d \rhod \phi d \theta }[/tex]

[tex]\mathbf{\implies \int ^{2 \pi}_{0} \int ^{\dfrac{\pi}{3}}_{0} \ sin^3 \phi cos^2 \phi sin^2 \theta \ \int^1_0 \ \rho^6 d \rho d \phi d \theta }[/tex]

[tex]\mathbf{\implies \int ^{2 \pi}_{0} \int ^{\dfrac{\pi}{3}}_{0} \ sin^3 \phi cos^2 \phi sin^2 \theta \Big [ \dfrac{\rho^7}{7} \Big]^1_0 \ d \phi d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta \int ^{\dfrac{\pi}{3}}_{0} \ sin^3 \phi cos^2 \phi \ d \phi d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta \int ^{\dfrac{\pi}{3}}_{0} \ sin \phi( 1- cos^2 \phi)cos^2 \phi \ d \phi d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{cos^5 \phi}{5}- \dfrac{cos ^3 \phi}{3} \Bigg ] ^{\dfrac{\pi}{3}}_{0} d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{cos^5 \dfrac{\pi}{3}}{5}- \dfrac{cos ^3 \dfrac{\pi}{3}}{3}- \dfrac{cos^5 0}{5}+ \dfrac{cos^3 0}{3} \Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{2}{15}- \dfrac{17}{480} \Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{(480\times 2) -(15\times 17)}{15\times 480}\Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{705}{15\times 480}\Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \int ^{2 \pi}_{0} 2sin ^2 d \theta}[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \int ^{2 \pi}_{0} (1-cos 2\theta) \ d \theta}[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \Bigg [\theta - \dfrac{sin 2 \theta }{2}\Bigg] ^{2 \pi}_{0}}[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \Bigg [2 \pi\Bigg] }[/tex]

[tex]\mathbf{\iiint_E y^2z^2 dV = \dfrac{47 \times 2 \pi}{6720}} }[/tex]

[tex]\mathbf{\iiint_E y^2z^2 dV = \dfrac{47 \pi}{3360} }[/tex]

Learn more about triple integral here:

https://brainly.com/question/2289273?referrer=searchResults

Answer:

Check out lesson 3.09, It'll help ;)

The lion runs 50 miles per hour which is 20 more than a house cat. 20 mph + house cat mph = 50 → house cat mph = 30mph

You just have to calculate the difference.

50-20=c

c=30