5. There are 1,000 meters in 1 kilometer.
You walk back and forth to school
every day. The school is 1.25 km from
your home. What is the distance you
walk, in meters, every day?

This is how the sum of 1/2+1/6+1/12+1/20 can be written in sigma notation.​

Answer:

It's A and C.. 0.40 is 40% as a decimal

The maximum and minimum values of f(x, y, z) = xyz subject to the constraint x^2 + y^2 + z^2 = 3 are 1 and -1.

To find the maximum and minimum values of f(x, y, z) = xyz subject to the constraint x^2 + y^2 + z^2 = 3, we can use the method of Lagrange multipliers.

Define the Lagrangian function L(x, y, z, λ) as follows:

L(x, y, z, λ) = xyz - λ(x^2 + y^2 + z^2 - 3)

Take partial derivatives of L with respect to x, y, z, and λ, and set them equal to 0:

∂L/∂x = yz - 2λx = 0

∂L/∂y = xz - 2λy = 0

∂L/∂z = xy - 2λz = 0

∂L/∂λ = -(x^2 + y^2 + z^2 - 3) = 0

Solve the system of equations formed by the partial derivatives to find the critical points.

From the first equation, we have yz = 2λx. Similarly, from the second and third equations, we have xz = 2λy and xy = 2λz.

Multiplying these equations together, we get:

xyz^2 = (2λx)(2λy)(2λz) = 8λ^3xyz

Since xyz ≠ 0 (as the constraint implies x, y, and z are not all zero), we can divide both sides by xyz to get:

z = 8λ^3

Similarly, we can find that x = 8λ^3 and y = 8λ^3.

Substituting these values into the constraint x^2 + y^2 + z^2 = 3, we get:

(8λ^3)^2 + (8λ^3)^2 + (8λ^3)^2 = 3

192λ^6 = 3

λ^6 = 3/192

λ^6 = 1/64

Taking the sixth root of both sides, we find:

λ = ±1/2

Substitute the values of λ into the equations x = 8λ^3, y = 8λ^3, and z = 8λ^3 to find the critical points.

For λ = 1/2:

x = 8(1/2)^3 = 1

y = 8(1/2)^3 = 1

z = 8(1/2)^3 = 1

For λ = -1/2:

x = 8(-1/2)^3 = -1

y = 8(-1/2)^3 = -1

z = 8(-1/2)^3 = -1

Evaluate the function f(x, y, z) = xyz at the critical points to find the maximum and minimum values.

For the critical point (1, 1, 1):

f(1, 1, 1) = 1 * 1 * 1 = 1

For the critical point (-1, -1, -1):

f(-1, -1, -1) = -1 * -1 * -1 = -1

Therefore, the maximum and minimum values of f(x, y, z)

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Ivan has created 9 sculptures.

We know, ratio is a comparison between two quantities or numbers. It expresses how many times one quantity is contained in another or how the two quantities relate to each other.

To find the number of sculptures he has created, we need to divide the number of paintings by the ratio of paintings to sculptures.

Number of paintings: 54

Ratio of paintings to sculptures: 6 paintings : 1 sculpture

Number of sculptures

= Number of paintings / Ratio of paintings to sculptures

= 54 / (6/1)

= 54 / 6

= 9

Therefore, Ivan has created 9 sculptures.

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a) The number of insects at t=0 days is 63.9

b) The growth rate is 0.071, or 7.1% per day.

c) The population after 10 days is 63.9.

d) The insect population will reach 13,507 insects approximately 26.4 days after the initial time.

e) The insect population will double approximately 19.0 days after the initial time.

In this problem, we are given a function that describes the size of a certain insect population at time t. We will use this information to answer various questions about the population, such as the initial size, growth rate, population after a certain number of days, and the time it takes for the population to reach specific thresholds.

The given function that describes the size of the insect population is

P(t) = 900 * 0.071.

(a) Determine the number of insects at t=0 days:

To find the number of insects at t=0 days, we substitute t=0 into the function P(t):

P(0) = 900 * 0.071 = 63.9

Therefore, the number of insects at t=0 days is 63.9.

(b) What is the growth rate of the insect population:

The growth rate of the insect population can be determined by examining the coefficient multiplying t in the function P(t).

In this case, the coefficient is 0.071. Since this coefficient is positive, it indicates that the population is growing.

The growth rate is 0.071, or 7.1% per day.

(c) What is the population after 10 days:

To find the population after 10 days, we substitute t=10 into the function P(t):

P(10) = 900 * 0.071 = 63.9

Therefore, the population after 10 days is 63.9.

(d) When will the insect population reach 13,507 insects:

To determine when the population reaches 13,507 insects, we need to solve the equation P(t) = 13,507. Let's set up the equation:

900 * 0.071 * t = 13,507

t = 13,507 / (900 * 0.071)

Simplifying the expression, we find:

t ≈ 26.4

Therefore, the insect population will reach 13,507 insects approximately 26.4 days after the initial time.

(e) When will the insect population double:

To determine when the insect population will double, we need to find the time at which P(t) is twice the initial population, P(0). Let's set up the equation:

2 * P(0) = 900 * 0.071 * t

2 * 63.9 = 900 * 0.071 * t

Simplifying the equation, we find:

t ≈ 19.0

Therefore, the insect population will double approximately 19.0 days after the initial time.

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Complete Question:

A The size of a certain insect population at time t (in days) obeys the function P(t) = 900 0.071

(a) Determine the number of insects at t=0 days.

(b) What is the growth rate of the insect population?

(c) What is the population after 10 days?

(d) When will the insect population reach 13507 insects ?

(e) When will the insect population double?

(Do not round until the final answer. Then round to the nearest tenth as needed.)

Answer:

Step-by-step explanation:

Kennel A:  c = 25d

Kennel B:  c = 30 + 20d

The values of the variables, a, b, and c obtained from the equation of the circle and the coordinates of the point P are;

a) a = 2

b = -2

c = 4

The general equation of a circle is; (x - h)² + (y - k)² = r²

Where;

(h, k) = The coordinates of the center of the circle

r = The coordinates of the radius of the circle

The specified equation of a circle is; x² + y² = 9

The coordinates of the center of the circle, is therefore, O = (0, 0)

a) The coordinates of the points P  and O indicates that the gradient of OP, obtained using the slope formula is; ((3·√3)/4 - 0)/(3/2 - 0) = ((3·√3)/4)/(3/2)

((3·√3)/4)/(3/2) = (√3)/2

The specified form of the gradient is; (√3)/a, therefore;

(√3)/a = (√3)/2

a = 2

The value of a is 2

b) The gradient of the tangent to a line that has a gradient of m is -1/m

The gradient of OP is; (√3)/2, therefore, the gradient of the tangent at P is -2/(√3)

The form of the gradient of the tangent at P is b/(√3), therefore;

-2/(√3) = b/(√3)

b = -2

The value of b is; -2

c) The coordinate of the point on the tangent, (0, (7·√3)/c) indicates

Slope of the tangent = -2/(√3)

((7·√3)/c - ((3·√3)/4))/(0 - (3/2)) = -2/(√3)

((7·√3)/c - ((3·√3)/4)) = (3/2) × 2/(√3) = √3

(7·√3)/c = √3 + ((3·√3)/4) = 7·√3/4

Therefore; c = 4

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Answer:

8.7

Leg

Step-by-step explanation:

Answer:

From left to right, top to bottom: 1 and 2 are neither, 1 and 2 are neither, 1 and 2 are neither, 1 and 2 are cooresponding angles, 1 and 2 are alterante interior angles.

Step-by-step explanation:

To solve the system of equations using substitution we replace the value of y from the first equation into the second equation to find the value of x.

solve the equation for x

using the value of x in either equation find the value of y

The solution for the system of equations is:

X= -1,5

y= 0.75

Answer:

Read carefully below

Step-by-step explanation:

In general, the inverse of a function f(x) is a function g(x) such that f(g(x)) = g(f(x)) = x for all values of x in the domain of f(x) and g(x). To find the inverse of a function, we need to switch the input and output variables and solve for the new output variable.

In this case, we are given the function -2f(x+5)-4. To find the inverse of this function, we need to switch the input and output variables and solve for the new output variable, which we will call y. We can do this by first moving all the terms to one side of the equation, and then solving for y:

-2f(x+5) - 4 = y

-2f(x+5) = y + 4

f(x+5) = (y + 4) / -2

This is the inverse function of -2f(x+5)-4. However, keep in mind that not all functions have inverses. In particular, a function must have the property that it is one-to-one (i.e. it maps unique inputs to unique outputs) in order to have an inverse. It is not clear from the given information whether the function -2f(x+5)-4 is one-to-one, so it is possible that it does not have an inverse.

Answer:

2/3

Step-by-step explanation:

Simplify

\(26=6\cdot \left(5-a\right)\\26=6\cdot 5+6\cdot -a\\26=30+6\cdot -a\\26=30+\left(6\cdot -1\right)a\\26=30-6a\\30-6a=26\)

Group constants

\(30-6a=26\\30-6a-30=26-30\\-6a+30-30=26-30\\-6a=26-30\\-6a=-4\\\)

Isolate the term, a

\(-6a=-4\\\frac{-6a}{-6}=\frac{-4}{-6}\\\frac{6a}{6}=\frac{-4}{-6}\\a=\frac{-4}{-6}\\\\\frac{-a}{-b} = \frac{a}{b}\\\\a=\frac{4}{6}\\a=\frac{2\cdot 2}{3\cdot 2}\\a=\frac{2}{3}\)

Answer:

1x+46 is the answer

Step-by-step explanation:

=4(-8x + 5) – (-33x – 26)

opening brackets to simplify

=-32x+20+33x+26

=-32x+33x+20+26

=1x+46 is the answer

hope it will help :)

Answer:

x+46

Step-by-step explanation:

4(-8x + 5) – (-33x – 26) ​

The first step is to distribute and simplify.

4 distributed to -8x will be -32x, and same thing goes with 5, which is 20.

-

The first part of the equation is -32x+20

Now, this is where it gets confusing.

If there is a sign before a pair of parentheses, you always put 1. In this case, it's -1 distributed to -33x and -26.

So, the second part of the expression will be +33x+26

The whole expression is

-32x + 20 + 33x + 26

add like terms, -32x+33x= x.

20+26=46.

The answer is x+46.

If the probability of rain is P, because there are only two outcomes, we know that the probability that will not rain is 1 - P

Let's say that the probability of rain tomorrow is P. There are two possible events here:

Then the second event is the negation of the first (Thus the probability is the composition), and notice that one of these events will happen, then the sum of the probabilities must be 1, then the probability that tomorrow will not rain is:

probability of not rain = 1 - P

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Step-by-step explanation:

y=-x since the gradient will be decreasing .

Answer:

B, 4

Step-by-step explanation:

To do this you can use the Pathagaryos thereom a^2+b^2=c^2

So we have (x-2)^2+x^2=20

We can expand (x-2)^2 and get

x^2-4x+4

so we have x^2 -4x +4 + x^2=20

We can combine like terms and subtract 20 from both sides to get

2x^2-4x-16=0

We can factor this and get (2x+4)(x-4)

which means that x=4 and -2

Obviously the length can't be a negative and so we reject x= -2 and we get x=4

We can double check it and see that

(4-2)^2+4^2=20

and it does

Answer: 100 cm

Step-by-step explanation:

1. Since the problem tells us that they are similar, we can infer that this larger rectangle is the same rectangle, only bigger.

2. With this information, we can do the equation 49/12.25 because then we will get the amount the rectangle grows by.

3. 49/12.25=4

4. Now, we can multiply the length(25) by 4.

5. 25 x 4=100

6. The length of the larger rectangle is 100.

I hope this helps!

Answer: 83r^2 - 59 + 60

Step-by-step explanation:

R^2 - 59R - 82r^2 + 60

= 83r^2 - 59 + 60

Two unit vectors orthogonal to both (8, 7, 1) and (-1, 1, 0) are (-1, -1, 15)/sqrt(227) and (-1, -1, 15)/sqrt(227).

To find two unit vectors orthogonal (perpendicular) to both (8, 7, 1) and (-1, 1, 0), we can use the cross product of the two given vectors. The cross product of two vectors will yield a vector that is orthogonal to both of them.

Let's calculate the cross product:

(8, 7, 1) × (-1, 1, 0) = [(7 * 0) - (1 * 1), (1 * -1) - (0 * 8), (8 * 1) - (7 * -1)]

= [-1, -1, 15]

Now, to obtain unit vectors, we divide this vector by its magnitude:

Magnitude of [-1, -1, 15] = sqrt((-1)^2 + (-1)^2 + 15^2) = sqrt(1 + 1 + 225) = sqrt(227)

Unit vector 1: (-1, -1, 15) / sqrt(227)

Unit vector 2: (-1, -1, 15) / sqrt(227)

Both of these unit vectors are orthogonal to both (8, 7, 1) and (-1, 1, 0).

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The amount of material needed = the surface area of the cylindrical can which is approximately calculated as: 332 square centimeters.

The material used = surface area of cylindrical can = 2πr(h + r).

Given the dimensions of the cylindrical can, we have:

radius (r) = 7/2 = 3.5 cm

height of cylindrical can (h) = 11.6 cm

π = 3.14

Amount of tin used = surface area = 2 * 3.14 * 3.5 * (11.6 + 3.5)

≈ 332 square centimeters

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Answer:

\(=x(2y-x)(2y-3x)\)

Step-by-step explanation:

We want to factor the following expression:

\(x(2y-x)^2+2x^2(x-2y)\)

First, notice that both of the terms have an x. Thus, let's factor out the x first:

\(=x((2y-x)^2+2x(x-2y))\)

Now, notice the similarity between (2y-x) and (x-2y). If we multiply either of them by a negative, they will be the same.

Therefore, we can factor out a negative from (x-2y). This will give us:

\(=x((2y-x)^2+2x(-(2y-x))\)

Since multiplication is commutative:

\(=x((2y-x)^2-2x(2y-x))\)

Now, we can factor out the (2y-x):

\(=x(2y-x)((2y-x)-2x)\)

And we can simplify this to get:

\(=x(2y-x)(2y-3x)\)

Alternate Method:

Starting from here:

\(x((2y-x)^2+2x(x-2y))\)

We can factor out a negative from the (2y-x)² term. This yields:

\(=x((-1(x-2y))^2+2x(x-2y))\)

Since -1 squared is positive, we can simplify:

\(=x((x-2y)^2+2x(x-2y))\)

Now, we can factor out the (x-2y):

\(=x(x-2y)((x-2y)+2x)\)

Simplifying gives:

\(=x(x-2y)(3x-2y)\)

While this looks different, they are exactly the same. If we factor out a negative from both the second and third term, we get:

\(=x(-(2y-x))(-(2y-3x))\)

The negatives will cancel, leaving us with:

\(=x(2y-x)(2y-3x)\)

This is exactly the same as what we acquired previously.

Therefore, both answers are correct.

Answer:

Step-by-step explanation:

Answer:

4,923.75ft²

Step-by-step explanation:

The area of the backyard = Length * width \

Area of the backyard = 75 3/4 * 81 1/4

Area of the backyard = 303/4 * 325/4

Area of the backyard = 6,154.6875ft²

If 3/4 of the fence is finished, the area of the finished part is expressed as;

Finished part = 1/5 * 6,154.6875

Finished part = 1,230.9375ft²

Part left unfinished = 6,154.6875-1,230.9375

Part left unfinished  = 4,923.75ft²

Answer: 235.5

Step-by-step explanation:

75 3/4= 75.75 (L). 81 1/4= 81.25 (W)

Perimeter= 314 3/4 of 314 is 235.5

The Delhi metro track increased by 20% in the last two years.

The Delhi metro track increased from 200 km to 240 km in the last two years. To calculate the percentage increase, we need to find the difference between the new and old lengths, divide it by the old length, and then multiply by 100.
The difference in length is 240 km - 200 km = 40 km.
The percentage increase, we divide the difference by the old length: 40 km ÷ 200 km = 0.2.
Finally, we multiply by 100 to get the percentage: 0.2 × 100 = 20%.
Therefore, the Delhi metro track increased by 20% in the last two years.
Another way to calculate the percentage increase is by using the formula:
Percentage increase = (new value - old value) ÷ old value × 100.
In this case, the new value is 240 km and the old value is 200 km.
So, the percentage increase = (240 km - 200 km) ÷ 200 km × 100 = 40 km ÷ 200 km × 100 = 0.2 × 100 = 20%.
Therefore, the Delhi metro track increased by 20% in the last two years.

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The BSA or Body surface area can be calculated by the formula is as follows: Body Surface Area= 0.007184 x (Height(m)^0.725) x (Weight(kg)^0.425)

Body surface area was developed as a metric to be used as such in the modulation of many various pharmaceutical treatments and as a standard index to various physiological measurements like glomerular filtration rate and cardiac output.

The Du Bois and its formula is one of the most widely used methods to determine a person's body surface area, in despite the fact that there are actually numerous variations on this concept. In the measurement of physiological and pharmacological data, the body surface area metric is much highly valued. The most typical reference point for chemotherapeutic drug dosing is body surface area.

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Step-by-step explanation:

does that mean you understand all the other questions but not 5. ?

that is strange, as this seems to be the easiest one of all the 6 questions.

to be sure, let's have a look at all 6.

1.

the distance MJ = 18 - 2 = 16

MX / XJ = 3/1

the ratio talkes about 3+1 = 4 basic units.

as the whole distance is 16, one such basic ratio unit is 16/4 = 4.

that means from M to X we have 3 of these basic ratio units, and from X to J the remaining 1 of the basic ratio units.

so, MX = 3×4 = 12

since M = 2, X = 2 + 12 = 14

2.

similar to 1.

just now

MY / YJ = 2/3

so, the ratio uses 2+3 = 5 basic units.

that means 1 basic unit is 16/5 = 3.2

2 basic units = 2 × 3.2 = 6.4

3 basic units = 3 × 3.2 = 9.6

so, MY = 6.4

since M = 2, Y = 2 + 6.4 = 8.4

3.

the distance A F = 5 - -7 = 5 + 7 = 12

AW / WF = 1/3

so, the ratio uses 1+3 = 4 basic units.

1 basic unit is 12/4 = 3

3 basic units is 3 × 3 = 9

so, AW = 3

since A = -7, W = -7 + 3 = -4

4.

the distance BF = 5 - -5 = 5 + 5 = 10

BX / XF = 3/2

so, the ratio uses 3+2 = 5 basic units.

1 basic unit = 10/5 = 2

2 basic units = 2 × 2 = 4

3 basic units = 3 × 2 = 6

so, BX = 6

since B = -5, X = -5 + 6 = 1

5.

the distance CE = 2 - -4 = 2 + 4 = 6

CY / YE = 1/1

so, the ratio uses 1+1 = 2 basic units.

1 basic unit = 6/2 = 3

so, CY = 3

since C = -4. Y = -4 + 3 = -1

6.

attention : this goes in the other direction than the other questions. all the others were looking from a point on the left to a point on the right.

but this one looks from a point on the right to a point on the left.

but the same principles apply.

the distance FD = |1 - 5| = |-4| = 4

a distance is always positive, so for this direction we need to apply the absolute value (formality we did that also for the other direction in the other questions, but there it was not necessary to use it explicitly) .

FZ / ZD = 5/3

so, the ratio uses 5+3 = 8 basic units.

1 basic unit = 4/8 = 0.5

3 basic units = 3 × 0.5 = 1.5

5 basic units = 5 × 0.5 = 2.5

so, FZ = 2.5

since F = 5, Z = 5 - 2.5 = 2.5

this is where the other direction really hits : since we have to go to the smaller side, we have to subtract the distance from the original coordinate (instead of the usual addition).

In order to have length twice width the answer should be 28 feet.

Let x be the width of the garden.

The length of the garden is 2x.

The perimeter of a rectangle is equal to 2(length + width)

So 42 = 2(2x + x)

42 = 2(3x)

x = 14

The width of the garden is 14 feet and the length is 2x = 2(14) = 28 feet.

The answer is 28 feet (Option a).

Mensuration is the branch of mathematics that deals with the measurement of geometric shapes and figures, such as length, area, and volume. It typically includes the study of geometric shapes, such as circles, triangles, and polygons, and their properties, such as perimeter, area, and volume. Mensuration is used in many fields, including engineering, physics, and computer graphics.

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