For the following system.
kx + y + z = 1
x + ky + z = 1
x+ y + kz = 1
Determine for what values of k the system has:
a) No solutions
b) One solution
c) A lot of solutions ...?

Using Snell’s law ( n1 sin θi = n2 sin θr ) at point A gives

1)        sin i1 = n sin r1             

2)        i1 = n r1

3)        i1 = α1 + β1

4)        r1 = β1 - γ

5)        α1 + β1 = n(β1 - γ)

6)        sin i2 = n sin r2

7)        i2 = n r2

8)        i2 = α2 + β2

9)        r2 = β2 + γ

10)      α2 + β2 = n(β2 + γ)

11)      α1 + α2 = (n - 1)(β1 + β2)

The lens maker's equation is derived by considering the image formed by the first refracting surface (left surface) and then using this image as the object for the second refracting surface. The equation takes into account the radii of curvature and the refractive indices of the lens and the surrounding medium.

The lens maker's equation is derived by considering the image formed by the first refracting surface (left surface) and then using this image as the object for the second refracting surface.

The equation takes into account the radii of curvature and the refractive indices of the lens and the surrounding medium.

The lens maker's equation can be stated as: 1/f = (n₂ - n₁)((1/R₁) - (1/R₂)), where f is the focal length, n₂ is the refractive index of the lens, n₁ is the refractive index of the surrounding medium (usually air), R₁ is the radius of curvature of the first surface of the lens, and R₂ is the radius of curvature of the second surface of the lens.

the answer is C (8,20)

Answer:

c) (8, 20)

Step-by-step explanation:

We must plug in the given value of x in the function and confirm that the value of y that is returned to us is the correct one.

a) (0, 18)

[tex]x=0,y=18[/tex]

[tex]y = 16 + 0.5(0)=16+0=16[/tex]

the values for y do not match, so it is not a solution.

b) (5, 19.5)

[tex]x=5,y=19.5[/tex]

[tex]y = 16 + 0.5(5)=16+2.5=18.5[/tex]

again, the values for y do not match so it is not a solution.

c) (8, 20)

[tex]x=8,y=20[/tex]

[tex]y = 16 + 0.5(8)=16+4=20[/tex]

For this option, the values of y do match, so (8, 20) are the missing values.

If EF = 6 and EG = 21 what is the value of FG?

The answer is 15.

The farmer owns 8 10/45 acres of land.

To find out how many acres of land the farmer owns, we need to multiply the number of sections by the size of each section. The size of each section is 2 1/5 acres.

The number of sections is 4 4/9. Using multiplication, we can calculate the total number of acres:

(2 1/5) x (4 4/9) = 10/5 x 40/9 = 400/45 = 8 10/45

Therefore, the farmer owns 8 10/45 acres of land.

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Final answer:

To find dy/dx by implicit differentiation of the equation xy=35, differentiate both sides with respect to x. Use the product rule to obtain x * (dy/dx) + y = 0 and solve for dy/dx, giving dy/dx = -y / x. Evaluate this at a specific point by substituting the x and y values from that point into the derived formula.

Explanation:

To find dy/dx by implicit differentiation of the equation xy = 35, we differentiate both sides of the equation with respect to x. Since y is a function of x, when we differentiate y we must apply the chain rule, resulting in the derivative y with respect to x, or dy/dx.

Starting with the original equation:

xy = 35

We apply differentiation to both sides with respect to x:

Using the product rule for the left side: d/dx (xy) = d/dx (35)

This gives us x * (dy/dx) + y * (1) = 0 because the derivative of a constant, like 35, is 0.

We then solve for dy/dx: dy/dx = -y / x.

To evaluate the derivative at a given point, you substitute the x and y coordinates of that point into the derivative equation. For example, if we are given a point (5,7), we substitute into the dy/dx equation to find the derivative at that point:

dy/dx = -y / x = -(7) / (5)

Therefore, dy/dx at the point (5,7) is -7/5 or -1.4.

24 is 30% of 80

Image provided.

Answer with explanation:

⇒Domain:

f(x)= -x² -2 x +15

   = - (x²+2 x -15)

Splitting the middle term

 = - (x²+5 x - 3 x -15)

= -[ x × (x+5) -3× (x+5)]

= -(x-3)(x+5)

y=f(x)=(3 -x)(x+5)

Domain of the function is defined as set of all values of x, for which y is defined.

f(x) is defined as all real values of x.So, Domain = R.

⇒Range:

[tex]y=-x^2-2 x +15\\\\y=-(x^2+2 x-15)\\\\ y=-[(x+1)^2-1-15]\\\\y= -(x+1)^2+16\\\\ 16 -y=(x+1)^2\\\\x+1=\pm\sqrt{16-y}\\\\x=\pm\sqrt{16-y}-1[/tex]

Range of the function is defined as set of all values of y, for which x is defined.

⇒16 -y ≥ 0

⇒y ≤ 16

Option B

The domain is all real numbers. The range is {y|y ≤ 16}.

True, the period of y = cos x occurs every 2pi units.

Equations using trigonometric functions that hold true for all possible values of the variables are known as trigonometric identities.

Given:

Trigonometric function,

y = cosx.

To find the period:

Use formula,

period = 2π/the coefficient of cosx.

Period = 2π

Therefore, the period of y = cosx is 2π.

To learn more about the trigonometric identities;

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Answer:

pb and 5

Step-by-step explanation:

Answer: Hello there!

I understand the notation 5 1/3 is equivalent to 5 + 1/3 = 5.333...

our equation is 5 +1/3 +(-3 +9/18)

first we could simplify the 9/18 as 1/2 (dividing both numerator and denominator by 9)

then the equation is:

=5+ 1/3 - (3 +1/2)

=5 + 1/3 - 3 - 1/2

=5 -3 + (1/3 - 1/2)

the common factor betwe 3 and 2 is 6, then:

=2 + (2/6 - 3/6) = 2 -1/6

this can be written as:

= 2 - 1/6 = 1 + 1 -1/6 = 1 + ( 1-1/6) = 1 + 5/6

or = 1 5/6 using the original notation.

To determine which system has no solutions, we analyze a set of inequalities to identify the contradictory statement.

To determine which system has no solutions, we need to analyze the given inequalities:

x < 5

x > 1

x < 5

x < 1

x > 5

x > 1

x > 5

x < 1

From the list, the system 'x < 1' and 'x > 1' has no solutions as it creates a contradictory statement.

Answer:

297 pages

Step-by-step explanation:

Number of pages of longest book = 396 pages

Now we are given that her shortest book has one-fourth as many pages as the longest.

Number of pages of shortest book = [tex]\frac{\text{Number of pages in longest book}}{4}[/tex]

                                            = [tex]\frac{396}{4}[/tex]

                                            = [tex]99[/tex]

Now we are given that the book in the middle of her shelf has three times the number of pages of the shortest book.

So, Number of pages in middle book:

=  [tex]3 \times \text{Number of pages of shortest book}[/tex]

=  [tex]3 \times 99[/tex]

=  [tex]297[/tex]

Hence the middle book have 297 pages.

The car will need approximately 8.76 more gallons of gas to complete the remaining 440 miles of the 650-mile trip, assuming it continues to get the same mileage rate of 26.25 miles per gallon.

To calculate the additional amount of gas needed to complete the 650-mile trip, given that the car used eight gallons to travel the first 210 miles, we need to find the car's mileage rate and then determine how much more gas is required for the remaining distance.

Step-by-Step Explanation

Therefore, the car will need approximately 8.76 more gallons of gas to complete the rest of the trip at the same rate.

To calculate the additional gas needed for the remainder of a 650-mile trip after using 8 gallons for the first 210 miles, first determine the gas mileage and then calculate the gas required for the remaining 440 miles. The car will need approximately 16.76 more gallons of gas to complete the trip.

To determine how much more gas you will need to complete your 650-mile trip, after having used 8 gallons for the first 210 miles, you need to calculate your car's gas mileage and then use that information to find the amount of gas needed for the remaining distance.

First, let's compute your car's mileage:

Now, let's subtract the miles you have already traveled from the total trip length to find the remaining miles:

Finally, we calculate the amount of gas needed for the remaining distance:

Therefore, you will need approximately 16.76 more gallons of gas to complete your trip, assuming the gas mileage remains consistent.

Final answer:

To solve for final velocity (Vf) in acceleration calculations, identify the initial velocity (Vo), acceleration (a), and time (t), then use the formula Vf = Vo + at to find Vf. Substitute the known values into the formula to calculate the final velocity.

Explanation:

To solve for final velocity (Vf) in the formula for acceleration, follow these steps:

Identify the initial velocity, Vo, and any other known variables such as acceleration (a) and time (t).

Determine which equation to use, using the relationship that acceleration is the change in velocity over time (Δv/Δt). The formula to calculate final velocity is v = Vo + at.

Substitute the known values into the equation to solve for Vf. For example, with an initial velocity (Vo) of 30.0 km/h (which converts to 8.333 m/s), an unknown final velocity (Vf), and a time (Δt) of 8.00 seconds, the change in velocity (Δv) would be Vf - Vo.

To find Vf, rearrange the equation: Vf = Vo + (a Δt). Insert the known values into the equation and calculate the result.

As an example, if the initial velocity Vo is 0 and the acceleration is 0.40 m/s² over a time of 100 seconds, then the final velocity Vf is calculated as:

Vf = Vo + at = 0 + (0.40 m/s²) (100 s) = 40 m/s.

This procedure will yield the final velocity of an object assuming constant acceleration over the given time period.