A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

Answer:

D= -16

E= 0

F= 0

Step-by-step explanation:

The given equation is [tex]x^{2} + y^{2} + Dx + Ey + F = 0[/tex]

It is also given that the circle passes through (0,0) (16,0) and (8,8).

Inserting (0,0) in the equation, it gives

[tex]0 + 0 + 0 + 0 + F = 0[/tex]

This gives F = 0 .

Now inserting (16,0) , it gives

[tex]16^{2} + 0^{2} + D(16) + E(0) + 0 = 0[/tex]

[tex]D(16) = -256[/tex]

[tex]D = \frac{-256}{16}[/tex]

D = -16

Now inserting (8,8) , it gives

[tex]8^{2} + 8^{2} + (-16)(8) + (E)(8) + 0 = 0[/tex]

[tex]-16 + E = -16[/tex]

E = 0

Thus the equation of circle is

[tex]x^{2} + y^{2} + (-16)x  = 0[/tex]

We can draw the following graph and thus verify that points (0,0) (8,8) and (16,0) lie on graph.

The equation of the circle passing through points (0, 0), (8, 8), and (16, 0) is x^2 + y^2 - 16x - 16y = 0. Solving the system of equations derived from substituting the given points into the circle equation confirms these coefficients for D and E.

To find the equation of the circle that passes through the points (0, 0), (8, 8), and (16, 0), we can use the standard form of a circle's equation:

x

2

+

y

2

+ D

x

+ E

y

+ F = 0

Because the circle passes through the origin (0,0), we know that F = 0. With the remaining points (8, 8) and (16, 0), we can substitute these coordinates into the equation to form a system of equations.

Using point (8,8), the equation becomes:

64 + 64 + 8D + 8E + F = 0

Using point (16,0), the equation becomes:

256 + 16D + F = 0

Since F = 0, the system of equations is:

Solving these equations, we get:

The equation of the circle is therefore:

To verify the result, plotting the points and the graph of the circle on a graphing utility should show that the points lie on the circumference of the circle.

Answer:

t=8

Step-by-step explanation:

Subtract 12t from both sides

(12t-12t) + 16 = (13t-12t) + 24

16 = 1t (or just t) + 24

Then subtract 16 from both sides, and bring "t" over the equals sign making it a negative integer.

-t = 8

t = 8

Answer:

Step-by-step explanation:

The right triangle has three sides which can be called legs. The legs are; shorter leg. Longer leg and hypotenuse

Let the longer leg be x

One leg of a right triangle is 4 mm shorter than the longer leg. This means

The shorter leg = x - 4

the hypotenuse is 4 mm longer than the longer leg. This means

The hypotenuse = x + 4

So the legs of the triangle are

Shorter leg or side = x-4

Longer leg or side = x

Hypotenuse = x + 4

The lengths of the sides of the triangle are 12 mm, 16 mm, and 20 mm.

Let's use variables to represent the lengths of the sides:

According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse:

a² + b² = c²

Plugging in the values, we have:

x² + (x + 4)² = (x + 8)²

Expanding and simplifying, we get:

x² + x² + 8x + 16 = x² + 16x + 64

Combining like terms, we get:

x² - 8x - 48 = 0

Factoring the quadratic equation, we find:

(x - 12)(x + 4) = 0

Therefore, x = 12 or x = -4. We discard the negative value, so the lengths of the sides of the triangle are:

Answer:

Optical axis

Step-by-step explanation:

Optical axis is the rotational symmetry axis of the surfaces.

A line with a certain degree of rotational symmetry is called as the optical axis in an optical system.

It is the straight line that passes through the geometric center of the lens and joins two curvature centers of its surfaces.

It is also called as the principal axis.

Answer:

Step-by-step explanation:

(a) The general term of an arithmetic sequence is ...

  an = a1 + d(n -1)

If we let the sequence of population numbers be modeled by this, and we use t for the number of years, we want n=1 for t=0, so n = t+1 and we have ...

  p(t) = 854 +3(t+1-1)

  p(t) = 854 +3t

__

(b) The general term of a geometric sequence is ...

  an = a1·r^(n-1)

were r is the common ratio. Here, the multiplier from one year to the next is 1+10% = 1.10. Again, n=t+1, so the population equation is ...

  p(t) = 427·1.10^(t+1-1)

  p(t) = 427·1.10^t

There are total of 32 pages in the complete book.

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

Given is that Tyler reads 2/15 of a book on Monday, 1/3 of it on Tuesday, 2/9 of it on Wednesday, and 3/4 of the remainder on Thursday. He still has 14 pages.

Let the total number of pages in the book will be [x]. Then, we can write -

{2x/15} + {x/3} + {2x/9} + {3x/4} = x + 14

x{2/15 + 1/3 + 2/9 + 3/4} - 14 = x

259x/180 - 14 = x

1.44x - 14 = x

0.44x = 14

x = (14/0.44)

x = 32 (approx.)

Therefore, there are total of 32 pages in the complete book.

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Answer:

No because the probability of consecutive shows starting late are not independent events

Step-by-step explanation:

Is good begin with the definition of independent events

When we say Independent Events we are refering to  events which occur with no dependency of other evnts. Basically when the occurrence of one event is not affected by another one.

When two events are independent P(A and B) = P(A)xP(B)

But for this case we can't multiply 0.97x0.97x0.97 in order to find the probability that 3 consecutive shows start on time, because the probability for shows starting late are not independent events, because if the second show is late, the probability that the next show would be late is higher. And for this reason we can use the independency concept here and the multiplication of probabilities in order to find the probability required.

Yes, you can multiply (0.97)(0.97)(0.97) because the events are independent.

Yes, you can multiply (0.97)(0.97)(0.97) to find the probability that three consecutive shows on this network start on time. This is because the events are independent; the outcome of one show starting on time does not affect the outcome of the others.

Therefore, the probability that three consecutive shows start on time is the product of the probabilities of each show starting on time:  P(all three shows start on time) = 0.97 * 0.97 * 0.97 = 0.912673.

The best statement that could be used describe a polygon is that it is a closed plane figure with three or more sides that are straight. Thus, the correct answer is A: a closed plane figure with three or more sides that are straight.

Hope this helps! :)

Final answer:

To find the probability that the first to arrive waits no longer than 5 minutes and the probability that the man arrives first, follow the provided detailed steps.

Explanation:

To find the probability that the first to arrive waits no longer than 5 minutes:

Man arrives first: 1/6

Woman arrives first: 1/4

Man and Woman arrive simultaneously within 5 minutes: 1/12

The probability that the man arrives first: 1/6

Answer:

.65

Step-by-step explanation:

Strategy:

In this situation it is much easier to calculate the probability of the event we are looking for (he hits at least one target) by calculating the probability of its complement (he misses every target), and subtracting from 1.

In other words, we can use this strategy:

P(at least one hit)=1-P(miss all 10)

Calculations:

P(at least one hit)

=1-P(miss all 10)

=1-(0.9)^10

≈1-0.349

≈0.65

Answer:

P(at least one hit)≈0.65

I hope this helps!!!

The probability that Brandon will hit at least one of them is 0.65 approx.

Binomial distributions consists of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as

[tex]X \sim B(n,p)[/tex]

The probability that out of n trials, there'd be x successes is given by

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]

For the considered case, as each hit is independent of each other, and there is either hit or not hit, so this situation can be modeled by binomial distribution.

For this case, we get:

Then, we get: [tex]X \sim B(n=10,p=0.1)[/tex]

The  probability that Brandon will hit at least one of them is written symbolically as:

[tex]P(X \geq 1)[/tex]

We can rewrite it as:

[tex]P(X \geq 1) = 1 - P(X < 1) = 1 - P(X = 0)[/tex]

Using the probability function of binomial distribution, we get:

[tex]P(X \geq 1) = 1 - \: ^{10}C_0(0.1)^0(1-0.1)^{10} = 1 -0.9^{10} \approx 0.65[/tex]

Thus, the probability that Brandon will hit at least one of them is 0.65 approx.

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Answer:

D

Step-by-step explanation:

a meter is 100 centimeters so the ratio of the real car to the model is 300 centimeters to 3 centimeters, or 100:1 so D

.---------. _

'-O------O--'

Answer:

[tex]a_{n}=2^{n}+3[/tex]

Step-by-step explanation:

The given sequence is not arithmetic, it's a geometric sequence, that means the sequence is obtain using powers. The faster way to find the answer is to try options that fist with a geometric sequence. If we try the last one, we'll find that's the answer.

We need to try for n=1, n=2, n=3, n=4 and n=5:

a_{1}=2^{1}+3=5

a_{2}=2^{2}+3=4+3=7

a_{3}=2^{3}+3=8+3=11

a_{4}=2^{4}+3=16+3=19

a_{5}=2^{5}+3=32+3=35

Therefore, the right answer is the last choice, because as you can observe, it fits perfectly with the given sequence.

Answer:

d is the correct answer

Answer:75

Step-by-step explanation:

Answer: 0.5

Step-by-step explanation:

For binary distribution with parameters p (probability of getting success in each trial) and n (Total trials) , we have

[tex]\sigma=\sqrt{np(1-p)}[/tex]

We are given that ,

Total batches of televisions : n=25

The probability of defects : p= 0.01

Here success is getting defective batch .

Then, the standard deviation for the number of defects per batch will be :-

[tex]\sigma=\sqrt{(25)(0.01)(1-0.01)}\\\\=\sqrt{(25)(0.01)(0.99)}\\\\=\sqrt{0.2475}\\\\=0.497493718553\approx0.5[/tex] [Rounedde to the nearest tenth.]

Therefore, the standard deviation for the number of defects per batch =0.5

Answer:

The mean of a binomial distribution is given  by  mean  = n x p where n = the number of items and p equals the probability of success.  Here we have:

mean =  1632 x 0.57  =   930.2

Step-by-step explanation:

The mean for a binomial substitution = n x p

Mean = 1632 x 0.57 = 930.24

The answer would be D.

Picture:

Multiply P(x) by X, then add those together:

0 x 0.42 = 0

1 x 0.12 = 0.12

2 x 0.34 = 0.68

3 x 0.05 = 0.15

4 x 0.07 = 0.28

Mean = 0 + 0.12 + 0.68 + 0.15 + 0.28 = 1.23

Answer:

A i think

Step-by-step explanation:

Answer:

  1.047 ft

Step-by-step explanation:

The length of an arc is given by ...

  s = rθ

where s is the arc length, r is the radius, and θ is the central angle in radians. Your arc subtends an angle of 12° = (12·π/180) = π/15 radians. The length of the arc is then ...

  s = (5 ft)(π/15) = π/3 ft ≈ 1.047 ft

The weight swings through a total angle of 12°, corresponding to an arc length of approximately 1.047 feet on the circumference of the circle with radius 5 feet.

To answer this question, we need to understand that the weight swings through an arc, and this arc is a part of the circumference of a circle. Given the length of string (5 feet) is the radius, and the weight swings through 6° on either side of the vertical, we can calculate the total arc length.

Firstly, you should know that the total angle a circle encompasses is 360°. So, the weight swings through a total angle of 6° x 2 = 12°.

Secondly, recall the formula for the circumference of a circle is 2πr, or in our case 2π x 5 feet. Now, to find the length of the arc corresponding to 12°, we will use the proportion of the swing angle to the total angle, i.e., (12/360) x (2π x 5 feet).

In this way, the length of the arc travelled is almost 1.047 feet.

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Answer:

0.093 ft/min

Step-by-step explanation:

V = 4/3 π r³

Take derivative with respect to time:

dV/dt = 4π r² dr/dt

Plug in values:

141 = 4π (11)² dr/dt

dr/dt = 141 / (484π)

dr/dt ≈ 0.093

The radius is increasing at 0.093 ft/min.

The increasing rate of the radius is equal to 0.093 ft/min.

Volume is defined as the space occupied by the object in a three-dimensional space. The sphere is the shape of a circular ball.

The volume is calculated by the formula below,

V = 4/3 π r³

Take the derivative with respect to time:

dV/dt = 4π r² dr/dt

Solve the equation for the rate of change of radius of the sphere,

141 = 4π (11)² dr/dt

dr/dt = 141 / (484π)

dr/dt ≈ 0.093

Therefore, the increasing rate of the radius is equal to 0.093 ft/min.

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Answer:

It would Cost $125 for a 60 day supply of cat food.

Step-by-step explanation:

Given:

Rover eats cat food each day = [tex]\frac{3}{4}[/tex] = 0.75 can

Bono eats cat food each day = [tex]\frac{1}{2}[/tex] = 0.5 can

Each day consumption for both cats = 0.75+0.5 = 1.25 can

Each day both cats consume = 1.25 cans

For 60 days both cats consume = Number of cans in 60 days.

By Using Unitary method we get;

Number of cans in 60 days = [tex]1.25\times60=75 \ cans[/tex]

Now Cans are sold in a pack of 3.

Hence we will divide number of cans in 60 days with 3 we get;

Number of can packs required for 60 days = [tex]\frac{75}{3}=25 \ can \ packs[/tex]

Now Cost for each Can packs(3 can) = $5.00

Hence Cost for 25 Can packs (75 cans) = Price for 25 can packs(75 can)

By Using Unitary method we get;

Price for 25 can packs (75 cans) = [tex]5\times 25 = \$125[/tex]

Hence Price for 25 can packs (75 cans) which are used for 60 supply of cat food is $125.

Answer:

The answer is 1.

Answer:

1

Step-by-step explanation:

any variable of power 0 equal to 1

Answer:

The total number of questions are 40

Step-by-step explanation:

let the total number of questions be "x".

percent of questions answered correctly are 80.

In fractions 80% is [tex]\frac{4}{5}[/tex]

The total number of correctly answered questions are 32.

the equation is,

[tex](\frac{4}{5})(x) = 32[/tex]

thus , x = [tex]\frac{(5)(32)}{4}[/tex]

x= 40

Answer:

  x³ -6x² +x +20

Step-by-step explanation:

The distributive property is useful for making sure you have all of the partial products.

  (x^2–x–4)(x–5) = x(x^2–x–4) -5(x^2–x–4)

  = (x^3 -x^2 -4x) +(-5x^2 +5x +20)

  = x^3 +(-1-5)x^2 +(-4+5)x +20

  = x^3 -6x^2 +x +20

Answer:

Spearman's correlation

Step-by-step explanation:

A researcher records each student’s rank in his/her high school graduating class and the student’s rank in the college graduating class.

The correlation that should be used to measure the relationship between these two variables is - Spearman's correlation

This correlation gives a statistical measure of similar relationship between paired data.

This is used to evaluate relationships involving ordinal variables.

Answer:

  $62,490.65

Step-by-step explanation:

If we assume her deposits are at the beginning of the month, and that the interest is compounded monthly, the future value is that of an "annuity due." The formula is ...

  FV = P(1+r/n)((1+r/n)^(nt)-1)/(r/n)

where r is the APR (.0276), n is the number of yearly compoundings (12), P is the monthly payment ($280), and t is the number of years (15). Putting the numbers into the formula and doing the arithmetic, we get ...

  FV = $280(1.0023)(1.0023^180 -1)/(.0023) ≈ $62,490.65

Angelica's account balance after 15 years will be $62,490.65.

_____

If her deposits are at the end of the month, the balance will be $62,347.25.

[tex] \frac{5}{3} = \frac{x}{7} [/tex]

you them cross multiply

5×7=35

3×X= 3x

35=3x

x(n)= 11.6

Answer:

n = 35/3 or 11 2/3.

Step-by-step explanation:

5/3 = n/7

Cross multiply:

3n = 5*7

3n = 35

n = 35/3

Answer:

Step-by-step explanation:

We want to find 95% confidence interval for the mean of the weight of of textbooks.

Number of samples. n = 28 textbooks weight

Mean, u =76 ounces

Standard deviation, s = 12.3 ounces

For a confidence level of 95%, the corresponding z value is 1.96. This is determined from the normal distribution table.

We will apply the formula

Confidence interval

= mean +/- z ×standard deviation/√n

It becomes

76 +/- 1.96 × 12.3/√28

= 76 +/- 1.96 × 2.3113

= 76 +/- 4.53

The lower end of the confidence interval is 76 - 4.53 =71.47

The upper end of the confidence interval is 76 + 4.53 = 80.53

Therefore, with 95% confidence interval, the mean textbook weight is between 71.47 ounces and 80.53 ounces

Answer:

The amount invested in bonds = 35,000

The amount invested in CD = 15,000.

Step-by-step explanation:

The total amount that Daniela invested is $50,000, this means if we call the amount invested in bonds [tex]b[/tex], and the amount invested in CD [tex]d[/tex], then we have:

[tex]b+d=50,000[/tex] this says the total amount Daniela invested is $50,000.

And since the amount invested in bonds [tex]b[/tex] is $5,000 more than twice the amount Daniela put into the CD, we have:

[tex]b=5,000+2d[/tex].

Thus, we have two equations and two unknowns [tex]b[/tex] and [tex]d[/tex]:

(1). [tex]b+d=50,000[/tex]

(2). [tex]b=5,000+2d[/tex],

and we solve this system by substituting [tex]b[/tex] from the second equation into the first:

[tex]b+d=50,000\\5,000+2d+d=50,000\\3d=45,000\\\\\boxed{d=15,000}[/tex]

or, the amount invested in CD is $15,000.

With the value of [tex]d[/tex] in hand, we now solve for [tex]b[/tex] from equation(2):

[tex]b=5,000+2d\\b=5000+2(15,000)\\\boxed{b=35,000}[/tex]

or, the amount invested in bonds is $35,000.

The ordered pair (0 , -2) lies on the line with equation 3 x - y = 2

Step-by-step explanation:

To prove that a point lies on a line

The equation of the line is 3 x - y = 2

a) Point (0 , -2)

∵ x = 0 and y = -2

- Substitute the values of x and y in the left hand side

∵ The left hand side is 3 x - y

∵ 3(0) - (-2) = 0 + 2 = 2

∴ The left hand side = 2

∵ The right hand side = 2

∴ The two sides of the equation are equal

∴ The ordered pair (0 , -2) lies on the line

b) Point (-3 , 4)

∵ x = -3 and y = 4

- Substitute the values of x and y in the left hand side

∵ The left hand side is 3 x - y

∵ 3(-3) - (4) = -9 - 4 = -13

∴ The left hand side = -13

∵ The right hand side = 2

∴ The two sides of the equation are not equal

∴ The ordered pair (-3 , 4) doesn't lie on the line

c) Point (1 , -5)

∵ x = 1 and y = -5

- Substitute the values of x and y in the left hand side

∵ The left hand side is 3 x - y

∵ 3(1) - (-5) = 3 + 5 = 8

∴ The left hand side = 8

∵ The right hand side = 2

∴ The two sides of the equation are not equal

∴ The ordered pair (1 , -5) doesn't lie on the line

The ordered pair (0 , -2) lies on the line with equation 3 x - y = 2

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Answer:

[tex]f(x)=3^{3x-1}[/tex].

The domain of the function is the set of all real number and the range is [tex](0,\infty)[/tex]

Step-by-step explanation:

Given:

The function is  given as:

[tex]f(x)=\frac{1}{3}(81)^{\frac{3x}{4}}[/tex]

Using the rule of the exponents, [tex]a^{mn}=(a^m)^n[/tex],

[tex]f(x)=\frac{1}{3}((81)^{\frac{1}{4}})^{(3x)}\\f(x)=\frac{1}{3}(\sqrt[4]{81} )^{3x}\\f(x)=\frac{1}{3}(3)^{3x}\\f(x)=\frac{3^{3x}}{3^1}[/tex]

Using the rule of the exponents,[tex]\frac{a^m}{a^n}=a^{m-n}[/tex],

[tex]f(x)=3^{3x-1}[/tex]

Therefore, the simplified form of the given function is:

[tex]f(x)=3^{3x-1}[/tex]

Key aspects:

The given function is an exponential function with a constant base 3.

Domain is the set of all possible values of [tex]x[/tex] for which the function is defined.

The domain of an exponential function is a set of all real values.

The range of an exponential function is always greater than zero.

Therefore, the domain of this function is also all real values and the range is from 0 to infinity.

Domain: [tex]x \epsilon (-\infty,\infty)[/tex]

Range: [tex]y\epsilon (0,\infty)[/tex]

Answer: 1/3 27 all real numbers y>0

Step-by-step explanation: