The acceleration due to gravity, g, is given by g=GMr2, where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant.
(a) Suppose that we increase from our distance from the center of the Earth by a distance Δr=x. Use a linear approximation to find an approximation to the resulting change in g, as a fraction of the original acceleration: Δg≈ g× (Your answer will be a function of x and r.)
(b) Is this change positive or negative? Δg is (Think about what this tells you about the acceleration due to gravity.)
(c) What is the percentage change in g when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.29 km; assume the radius of the Earth is 6400 km)? percent change =1

Answer:

Step-by-step explanation:

You have to assume the the marked lines are parallel.

Alternate exterior angles are congruent, so ...

  (15x -26)° = (12x +1)°

  3x = 27 . . . . . . divide by °, add 26-12x

  x = 9 . . . . . . . . .divide by 3

Then the alternate exterior angles are ...

  (12·9 +1)° = 109°

___

The angle at upper right is vertical with the obtuse interior angle of the triangle whose other interior angles are marked. The sum of them is 180°, so we have ...

  28° +109° +(4y -9)° = 180°

  4y = 52 . . . . . . divide by °, subtract 128

  y = 13 . . . . . . . . divide by 4

Then the third interior angle of the triangle is ...

  (4·13 -9)° = 43°

Answer(s):

d= 4 (D=4) or 4=d (4=D)

Step-by-step explanation: You can solve it in 2 ways.

The [tex]1^{st}[/tex] way:

4d−4 = 5d−8

   +4        +4

   4d = 5d-4

  -5d   -5d

    -1d = -4

     /-1    /-1

      d = 4

The [tex]2^{nd}[/tex]

4d−4 = 5d−8

   +8        +8

   4d+4 = 5d

  -4d       -4d

        4 = 1d

        /1    /1

        4 = d

Hope this helps. :)

Answer:

the 3rd one (:

Step-by-step explanation:

Answer:

First One:

x { 6, 9, 12, 15, 18 }
y { -14, -4, 8, 22, 38 }

Step-by-step explanation:

Not sure if the other guy is trolling...
The x values constantly rise by 3, so we find the y value change
The differences are 10, 12, 14, 16
If we do it again we get 2, 2, 2. So since we did the reduction 2 times, this is a quadratic function.
Why isn't "the third one"
Constant rise by 2

Change of 9, 9, 9, 9
This is one reduction, so it is not quadratic.

Answer:

Step-by-step explanation:

Alice borrowed 16700 from the bank at a simple interest rate of 9% to purchase a used car. It means that the interest is not compounded. Simple interest is usually expressed per annum. The formula for simple is

I = PRT/100

Where

I = interest

P = principal(amount borrowed from the bank)

R = 9% ( rate at which the interest is charged

T = number of years

At the end of the loan,she had paid a total of 24215. This means that the interest + the principal = 24215

Therefore,

The interest = 24215 - 16700 = 7515

Therefore

7517 = (16700 × 9 × t)/100

751700 = 150300t

T = 751700/150300

T = 5 years

Converting 5 years to months,

1 year 12 months

5 months = 12 × 5 = 60 months

Final answer:

Alice's car loan was for 60 months. This was calculated using the simple interest formula and the total interest paid, which was the difference between the total amount paid and the principal amount borrowed.

Explanation:

To determine how many months the car loan was, we can use the simple interest formula, I = PRT, where:

Alice borrowed $16,700 at a rate of 9%, and the total amount paid was $24,215. The total interest paid is the total amount minus the principal, which is $24,215 - $16,700 = $7,515.

Using the formula, we get:

Thus, Alice's car loan was for 60 months.

Answer: The company has to record a loss of $2,000

Step-by-step explanation:

The accounting for bonds retired early would require the company to pay out cash to remove the bonds payable from its balance sheet. To determine the gain or loss, if the cash paid is less than the carrying value of the bond, then a gain is determined, if the cash paid is more than the carrying value of the bond, then a loss is determined.

From the question the company pays $100,000 which is more than the carrying value of $98,000. Therefore, the company would record a loss $2,000.

Answer:

[tex]\frac{(2-y)x}{5}\text{ litres}[/tex]

Step-by-step explanation:

Let l litre of 7 grams of salt per litre is combined with x litres of water which contains y grams of salt per litre to yield a solution with 2 grams of salt per litre.

Thus,

Salt in l litre + salt in x litre = salt in resultant mixture

7l + xy = 2(l +x)

7l + xy = 2l + 2x

7l - 2l = 2x - xy

5l = x(2-y)

[tex]\implies l =\frac{x(2-y)}{5}[/tex]

Hence, [tex]\frac{x(2-y)}{5}[/tex] litres of 7 gram of salt per litre is mixed.

Answer:

Step-by-step explanation:

Having drawn the line, Kendall must verify that the point P belongs to the line y = 2x-1 and then calculate the distance between A-P  and verify if it is the closest to A or there is another one of the line

Having the point P(3,5) substitue x to verify y

y=2*(3)-1=6-1=5 (3,5)

Now if the angle formed by A and P is 90º it means that it is the closest point, otherwise that point must be found

[tex]d_{AP}=\sqrt{(y_{2}-y_{1})^{2}+(x_{2}-x_{1})^{2}}=\sqrt{(5-7)^{2}+(3-(-2}))^{2}}=\\\sqrt{(-2)^{2}+(5)^{2}}=\sqrt{29}[/tex]

and we found the distance PQ and QA

; [tex]d_{PQ}=\sqrt{125}[/tex], [tex]d_{QA}=12[/tex]

be the APQ triangle we must find <APQ through the cosine law (graph 2).

Answer:

It's D

Step-by-step explanation:

Hope this helps :))

Answer:

g equals 1

g = 1

Isolate the variable by dividing each side by factors that don't contain the variable.

Hope this helps!

Answer:

g=1

Step-by-step explanation:

Given equation is \[−3+5+6g=11−3g\]

Simplifying, \[2 + 6g = 11 - 3g\]

Bringing all terms containing g to the left side of the equation and all the numeric terms to the other side,

\[3g + 6g = 11 - 2\]

=> \[9g = 9\]

=> \[g=\frac{9}{9}\]

=> g = 1

Validating by substituting in the given equation:

Left Hand Side = -3 + 5 + 6 = 8

Right Hand Side = 11 - 3 = 8

Hence the two sides of the equation are equal when g = 1.

Answer:

$1.00

Step-by-step explanation:

Total Budget = $35

Cake Spent = $18

Remaining Money = 35 - 18 = $17

Since, each balloon is worth $4, to find how many balloons he will get with remaining money ($17), we divide the the remaining money by price of each balloon:

17/4 = 4.25

We can't have fractional balloons, so Abit can get 4 balloons, MAXIMUM.

4 balloons cost = $4 per balloon * 4 = $16

So, from $17 if he spends $16 for balloons, he will have left:

$17 - $16 = $1.00

Answer:

8923 years

Step-by-step explanation:

Half life of C-14 = 5730yrs

decay rate= 34%

Halt life (t^1/2) = (ln2) / k

5730 = (ln2) /k

k = (ln2) / 5730

k = 1.209 * 10^-4

For first order reaction in radioactivity,

ln(initial amount) = -kt

ln(34/100) = -(1.209*10^-4)t

-1.0788 = -(1.209*10^-4)t

t = -1.7088/ -1.209*10^-4

t = 8923 years

It would take 8918 years for the tree to decay to 34%.

The half life is the time taken for a substance to decay to half of its value. It is given by:

[tex]N(t) = N_0(\frac{1}{2} )^\frac{t}{t_\frac{1}{2} } \\\\where\ t=period, N(t)=value\ after\ t\ years, N_o=original\ amount, t_\frac{1}{2} =half\ life\\\\Given\ t_\frac{1}{2} =5730,N(t)=0.34N_o, hence:\\\\0.34N_o=N_o(\frac{1}{2} )^\frac{t}{5730} \\\\t=8918\ years[/tex]

It would take 8918 years for the tree to decay to 34%.

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Answer:

10.44

Step-by-step explanation:

The weighted average cost per unit method seeks to get the cost of goods sold as an average of all cost of goods in the inventory as at the time of sales.

Part of its objective is to strike a balance between the (FIFO and LIFO) inventory valuation methods.

Beginning inventory ( Jan) = 10 units

Cost of beginning inventory per unit = $10

Total cost of beginning inventory = Cost * Number of units

In this case (10*$10) = $100

Additional purchase (Jan 5) = 8 units

Cost of additional purchase per unit = $11

Total cost of additional purchase = 8 * $11 = $88

Weighted average cost per unit at the time 11 units are sold on January 7 = Total cost of units at that time / number of units available at that time.

= ($100+ $88) / (10+8)

= 188/18

=10.44 (approximated to 2 decimal places)

I hope this helps make the concept clear.

Answer:

The population of this country in 11 years will be 76972702.82

Step-by-step explanation:

The population of this country in 11 years can be calculated using the formula

Population in 11 years =  Starting population x [tex](1+growth rate)^{period}[/tex]

Thus

Population in 11 years =50000000×[tex](1+0.04)^{11}[/tex] =76972702.82

The population of this country in 11 years will be 76.97 million.

In order to determine the population of a country in 11 years, this formula would be used:

FV = P (1 + r)^n

50 million x (1 + 0.04)^11

= 50 million x (1.04)^11

= 76.97 million

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Final answer:

Using the principle of inclusion-exclusion, it's calculated that 25% of the teachers surveyed had neither high blood pressure nor heart trouble.

Explanation:

To find the percentage of teachers who had neither high blood pressure nor heart trouble, we can use the principle of inclusion-exclusion in set theory. We begin by adding the number of teachers with each condition, then subtract those counted twice because they have both conditions.

The formula we will use is:

Total surveyed - (High blood pressure + Heart trouble - Both) = Neither condition

Substituting the numbers from the survey, we get:

120 - (70 + 40 - 20) = 120 - (90) = 30

So, 30 teachers had neither high blood pressure nor heart trouble.

To find the percentage, we divide the number of teachers with neither condition by the total surveyed and then multiply by 100:

(30 / 120) * 100 = 25%

Therefore, 25% of the teachers surveyed had neither high blood pressure nor heart trouble.

Answer:

[tex]x= +2\sqrt{5} or -2\sqrt{5}[/tex]

Step-by-step explanation:

for the given equation,

[tex]4 - x^{2} = -16[/tex]

using rules of equation and inverse operations as isolating x on one side of equation,

interchanging sides  of equation  we get,

[tex]x^{2} =20[/tex]

[tex]x= +\sqrt{20} or x= -\sqrt{20}[/tex]

[tex]x= +2\sqrt{5}  or x= -2\sqrt{5}[/tex]

Answer:

7 years old

Step-by-step explanation:

Robert's mother pours a cup of milk for him and realizes that cup has a small crack so she pour the milk into another cup. Robert is not upset because he knows that the amount of milk has remained same.

As Robert is 7 year old . He can groups the milk in the cup according to size , shape and color. He has a better understanding of numbers and can understand problems easily.

Robert's recognition of the conservation of volume implies he is at least 7 years old, the age at which children typically enter the concrete operational stage and understand this concept.

The question asks us to determine Robert's age based on his understanding of the concept of conservation of volume. According to developmental psychology, children gain the ability to understand that a change in the form of an object does not necessarily imply a change in the volume or quantity after a certain age. This concept is typically acquired during the concrete operational stage of development, which is from about 7 to 11 years old. Since Robert recognizes that the same amount of milk remains constant despite being poured into cups of different shapes, we can infer that Robert is at least in the concrete operational stage of development. Thus, Robert is at least 7 years old.

Answer: Choice B) -5.2 degrees

=====================

Work Shown:

Add up the given temperatures

-42 + (-17) + 14 + (-4) + 23 = -26

Then divide by 5 since there are 5 values we're given

-26/5 = -5.2

Answer:

The distance 5 miles North-East of the intersection between the car and the truck increasing at 71.06 miles per hour at that moment.

Step-by-step explanation:

Looking at the attached figures, Fig 1 shows the diagram of the car and the truck.

Using Pythagoras theorem on Fig 1a,

[tex]l^{2} = \sqrt{3^{2} + 4^{2} }[/tex]

[tex]l = \sqrt{9 +16} \\\\l= \sqrt{25} \\\\l = 5 miles[/tex]

The resultant displacement between the car and the truck at that same moment is 5 miles.

From the velocity vector diagram on Fig 2,

The resultant velocity R is given as

[tex]R = \sqrt{45^{2} + 55^{2} }\\\\R = \sqrt{2025 + 3025 }\\\\R = \sqrt{5050 }\\\\R = 71.06mph[/tex]

Therefore, the distance 5 miles North-East of the intersection between the car and the truck increasing at 71.06 miles per hour at that moment.

[tex]\boxed{a_{n}=-3+8(n-1)}[/tex]

The explicit formula for the general nth term of the arithmetic sequence is given by:

[tex]a_{n}=a_{1}+d(n-1) \\ \\ \\ Where: \\ \\ a_{n}:nth \ term \\ \\ n:Number \ of \ terms \\ \\ a_{1}:First \ term \\ \\ d:common \ difference[/tex]

Here we know that:

[tex]a_{1}=-3 \\ \\ a_{10}=69[/tex]

So, our goal is to find the common difference substituting into the formula:

[tex]a_{10}=a_{1}+d(10-1) \\ \\ 69=-3+d(9) \\ \\ Solving \ for \ d: \\ \\ 9d=69+3 \\ \\ 9d=72 \\ \\ d=8[/tex]

Finally, we can write the explicit formula as:

[tex]\boxed{a_{n}=-3+8(n-1)}[/tex]

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The explicit formula for the general nth term of the arithmetic sequence is an = 8n - 11.

To find the explicit formula for the general nth term of the arithmetic sequence with a1 = -3 and a10 = 69, we can use the formula for the nth term of an arithmetic sequence, which is:

an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.

First, we find the common difference by using the 10th term:

69 = -3 + (10 - 1)d

72 = 9d

d = 8

Now we have the common difference, so we can write the formula for the nth term:

an = -3 + (n - 1)(8)

Simplifying:

an = 8n - 11

This is the explicit formula for the general nth term of the given arithmetic sequence.

The projectile will be back at ground level at t = 0 (initially fired) and t = 8 seconds after being fired.

To find when the height of the projectile is less than 128 feet, we set s < 128 and solve for t using the given position equation:

[tex]\[ -16t^2 + v_0t + s_0 < 128 \][/tex]

Given:

- [tex]\( v_0 = 128 \)[/tex]  feet per second

- [tex]\( s_0 = 0 \)[/tex] (since the object is fired upward from ground level)

Substitute these values into the equation:

[tex]\[ -16t^2 + 128t < 128 \][/tex]

Now, let's solve this inequality for t. First, let's rewrite it in standard quadratic form:

[tex]\[ -16t^2 + 128t - 128 < 0 \][/tex]

Divide all terms by -16 to simplify:

[tex]\[ t^2 - 8t + 8 > 0 \][/tex]

Now, we need to find the values of t for which this inequality holds true.

To solve quadratic inequalities, we can find the roots of the corresponding quadratic equation [tex]\( t^2 - 8t + 8 = 0 \)[/tex], and then determine the intervals where the quadratic expression [tex]\( t^2 - 8t + 8 \)[/tex] is positive.

The roots of the quadratic equation [tex]\( t^2 - 8t + 8 = 0 \)[/tex] can be found using the quadratic formula:

[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

where [tex]\( a = 1 \), \( b = -8 \), and \( c = 8 \).[/tex]

[tex]\[ t = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \][/tex]

[tex]\[ t = \frac{8 \pm \sqrt{64 - 32}}{2} \][/tex]

[tex]\[ t = \frac{8 \pm \sqrt{32}}{2} \][/tex]

[tex]\[ t = \frac{8 \pm 4\sqrt{2}}{2} \][/tex]

[tex]\[ t = 4 \pm 2\sqrt{2} \][/tex]

So, the roots of the equation are [tex]\( t = 4 - 2\sqrt{2} \) and \( t = 4 + 2\sqrt{2} \).[/tex]

Now, we can test the intervals [tex]\( (-\infty, 4 - 2\sqrt{2}) \), \( (4 - 2\sqrt{2}, 4 + 2\sqrt{2}) \), and \( (4 + 2\sqrt{2}, \infty) \)[/tex] to determine where the inequality [tex]\( t^2 - 8t + 8 > 0 \)[/tex] holds true.

However, it seems there might be an error in the equation. Let's reassess the situation. If the projectile is fired straight upward from ground level, it will reach its maximum height and then fall back to the ground. The time at which it returns to the ground can be found by setting \( s = 0 \) in the position equation:

[tex]\[ -16t^2 + 128t = 0 \][/tex]

Factor out -16t:

-16t(t - 8) = 0

This equation will be true when either -16t = 0 or t - 8 = 0.

Solving each equation:

1.  -16t = 0

t = 0

2. t - 8 = 0

t = 8

So, the projectile will be back at ground level at t = 0 (when it's initially fired) and at t = 8 seconds after being fired.

The depth of the water in the cylindrical tank is increasing at a rate of 0.1 m/min. This result is gotten by using the concept of related rates in calculus and differentiating volume with respect to time. The given rate of water inflow and dimensions of the cylindrical tank are substituted into the resultant formula.

This question is dealing with rates of change, specifically in the context of volume and height within a cylindrical tank. It's a problem in calculus, more specifically related to related rates.

To start, let's understand that the volume V of a cylinder is given by the formula V = πr²h, where r is the radius, h is the height, and π is a constant (~3.14159). We can substitute r with 4m (half of the diameter of 8m) and differentiate both sides with respect to time. Differentiating both sides of the equation with respect to t (time) gives dV/dt =πr² dh/dt (since r is constant with respect to time, but h changes).

Given that the rate at which water flows into the tank, dV/dt, is 5m³/min we can substitute this into the formula. This gives us 5 = π(4)²dh/dt. Simplifying the equation gives us dh/dt = 5 / (π(4)²), which approximately equals 0.1 m/min when rounded to three decimal places.

So, the depth of the water in the tank is increasing at a rate of 0.1 m/min.

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The rate at which the depth of water is rising in the cylindrical tank with a diameter of 8 m and water flowing in at 5 m^3/min is approximately 0.199 m/min.

To find the rate at which the depth of the water is rising, we can use the formula for the volume of a cylinder:

where:

Differentiating both sides of the equation with respect to time \( t \) gives us:

Given that the diameter is 8 m, the radius \( r \) is 4 m. We are also given that the rate of change of volume dV/dt is 5 m³/min. Plugging in these values, we can solve for \( dh/dt \), the rate at which the depth of the water is rising:

Calculating this gives:

[tex]\[ dh/dt \approx 0.199 \text{ m/min} \][/tex]

So, the rate at which the depth of the water is rising is approximately 0.199 m/min.

Answer:

After 4 months

Step-by-step explanation:

The cost at both the programs consists of a "fixed cost" (reg fee)  & "variable cost" ( per month fee).

Let number of months be "x"

Yo-Yoga:

35 fixed

90 per month

So equation would be: 35 + 90x

Essence Yoga:

75 fixed

80 per month

So equation would be: 75 + 80x

To find number of month when cost would be same, we equate both equations and solve for x:

35 + 90x = 75 + 80x

10x = 40

x = 40/10

x = 4

hence, after 4 months, both cost would be same

Answer:

8x + 29 < 17

Step-by-step explanation:

Let the unknown number be x .

According to the question ,

8 times x when added to 29 will give a sum less than 17 .

So , we can write

8x + 29 < 17.

We can also solve it

so we will get

8x < -12

x<-1.5.

Answer:

Sometime between the third and fourth year (3.5 years)

Step-by-step explanation:

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

[tex]S=555+33y[/tex]

As for the French class, since we lose 2 students each year the equation is

[tex]F=230-2y[/tex]

We require to know the value of y in the exact moment when S=3F

[tex]555+33y=3(230-2y)[/tex]

Operating

[tex]555+33y=690-6y[/tex]

Reducing

[tex]39y=690-555[/tex]

[tex]39y=135[/tex]

[tex]y=135/39=3.5\ years[/tex]

It means that sometime between the third and fourth year, there will be 3 times as many students taking Spanish as French

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

Answer:

The probability of cured people in who took the remedy is 8/9.

Step-by-step explanation:

Success rate of the cold remedy = 88%

The number of people who took the remedy = 45

Now, 88% of 45 = [tex]\frac{88}{100}  \times 45 = 39.6[/tex]

and 39.6 ≈ 40

So, out of 45 people, the remedy worked on total 40 people.

Now, let E: Event of people being cured by cold remedy

Favorable outcomes = 40

[tex]\textrm{Probability of Event E}  = \frac{\textrm{Total number of favorable outcomes}}{\textrm{Total outcomes}}[/tex]

or, [tex]\textrm{Probability of people getting cured}  = \frac{\textrm{40}}{\textrm{45}}[/tex]  = [tex]\frac{8}{9}[/tex]

Hence, the probability of cured people in who took the remedy is 8/9.

Final answer:

To find the probability of at least four patients out of 25 actually having the flu, the binomial probability formula is used. For the expected number of flu cases, the probability (4%) is multiplied by the number of patients (25).

Explanation:

The question pertains to the calculation of probabilities regarding how many patients actually have the flu given a certain success rate of 4% for flu diagnosis among those reporting symptoms. Specifically, we are asked to find the probability that at least four out of the next 25 patients calling in actually have the flu. To solve this, we would use the binomial probability formula:

P(X ≥ k) = 1 - ∑ [ P(X = i) where i goes from 0 to k-1 ]

In this case, X represents the number of patients who actually have the flu, k is the number we are interested in (at least four), and P(X = i) is the probability that exactly i patients have the flu. To express the distribution of X, we use a binomial distribution as we are dealing with a fixed number of independent trials, each with two possible outcomes (having the flu or not).

To determine the expected number of patients with the flu, we would multiply the probability of an individual having the flu (4%) by the number of patients calling in, which is 25.

Expected number of patients with the flu = 25 * 0.04 = 1

Answer:

6 problems correct 7 incorrect, and 5 skipped over

Step-by-step explanation:

3+5=8 multiple of 8 that ends in 8:48

she got 6 problems corrects

4x6=48 48-13=35

35/5=7

20-7-8=5

Answer:

19.43 ft

Step-by-step explanation:

Using SOHCAHTOA,

opposite = x

Hypothenus = 25ft

Sin α = opposite /Hypothenus

Sin 51° = x/25

x = 25(Sin51°)

x = 19.4286

x = 19.43(approximate to 2 d.p)

Answer:

Option C

Step-by-step explanation:

As you can see in the graph the shaded region is in 1st quadrant which means that both x and y are greater than or less than equal to zero. From this statement only option A and B get eliminated. But still we will look further...

The equation of the yellow line should be determined in order to know the complete answer. If you don't know how to write equation of line in intercept form then just assume the line's equation to be :

y=mx + c    ; where m and c are constants.

Now you just need to find two points satisfying the line's equation. As you can see in the graph (2,0) and (0,-2) are the points lying on the line. Now put them in the assumed line's equation to determine the constants.

0 = m × 2 + c  .........equation (1)

-2 = m × 0 + c .........equation (2)

solving equation (2) gives c = -2

put the value of c in equation (1) then solve it to get the value of m

0 = m × 2 - 2

m=1

Therefore line's equation is y = x - 2

Since the value of y is greater than and equal to the value of y which we get from the line's equation so

y ≥ x - 2

rearrange the above inequality

x - y ≤ 2

So these three conditions are there

x ≥ 0

y ≥ 0

x - y ≤ 2

which gives the shaded region

Answer:

50 kg water.

Step-by-step explanation:

We have been given that the number of kilograms of water in a human body varies directly as the mass of the body.

We know that two directly proportional quantities are in form [tex]y=kx[/tex], where y varies directly with x and k is constant of variation.

We are told that an 87​-kg person contains 58 kg of water. We can represent this information in an equation as:

[tex]58=k\cdot 87[/tex]

Let us find the constant of variation as:

[tex]\frac{58}{87}=\frac{k\cdot 87}{87}[/tex]

[tex]\frac{29*2}{29*3}=k[/tex]

[tex]\frac{2}{3}=k[/tex]

The equation [tex]y=\frac{2}{3}x[/tex] represents the relation between water (y) in a human body with respect to mass of the body (x).

To find the amount of water in a 75-kg person, we will substitute [tex]x=75[/tex] in our given equation and solve for y.

[tex]y=\frac{2}{3}(75)[/tex]

[tex]y=2(25)[/tex]

[tex]y=50[/tex]

Therefore, there are 50 kg of water in a 75​-kg person.

To find out how many kilograms of water are in a 75 kg person, we can set up a proportion and solve for x. The number of kilograms of water is directly proportional to the mass of the body.

To solve this problem, we can use the concept of direct variation. Direct variation states that two variables are directly proportional to each other. In this case, the number of kilograms of water is directly proportional to the mass of the body.

We are given that an 87 kg person contains 58 kg of water. To find out how many kilograms of water are in a 75 kg person, we can set up a proportion:

(87 kg) / (58 kg) = (75 kg) / (x kg)

Cross multiplying, we get:

87 kg * x kg = 58 kg * 75 kg

Simplifying, we find that x = 39 kg.

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Answer:

[tex]V_{total} = \displaystyle\int_0^{60} A e^{-k t}~dt[/tex]

Step-by-step explanation:

We are given the following in the question:

Oil leaks out of a tanker at a rate of r = f(t) liters per minute, where t is in minutes.

[tex]f(t) = A e^{-k t}[/tex]

Let V be the volume, then we are given that rate of leakage is:

[tex]\displaystyle\frac{dV}{dt} = f(t) = A e^{-k t}[/tex]

Thus, we can write:

[tex]dV = f(t).dt = A e^{-k t}~dt[/tex]

We have to find the  a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

Thus, total amount of oil leaked will be the definite integral from 0 minutes to 60 minutes.

[tex]dV =A e^{-k t}~dt\\V_{total} = \displaystyle\int_a^b f(t)~dt\\\\a = 0\text{ minutes}\\b = 60\text{ minutes}\\\\V_{total} = \displaystyle\int_0^{60} A e^{-k t}~dt[/tex]

The units of integral will be liters.

The total amount of oil leaked in the first hour can be expressed as the definite integral [tex]\int_{0}^{60} A e^{-k t}[/tex] dt. The unit of this integral is liters, as the rate of oil leak is given in liters per minute and the time is in minutes.

The rate of the oil leak is given by [tex]f(t)=Ae^{-kt}[/tex], where A represents the initial rate of the leak, k is a constant, t is time in minutes, and e is the base of the natural logarithm. To express the total amount of oil leaked in the first hour in the form of a definite integral, we need to integrate this rate function from t=0 (the start of the leak) to t=60 (the end of the first hour).

So, the required integral would be expressed as: [tex]\int_{0}^{60} A e^{-k t} dt[/tex]. This integral calculates the total quantity of oil leaked in the first 60 minutes. The units of this integral would be liters, as the rate of the leak is given in liters per minute and the time is in minutes. Integrating a rate over time gives us a total quantity in the original unit of the rate, in this case liters.

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