Which group of values makes the expression 3x2y−2z UNDEFINED? A) x = –1; y = 0; z = 3 B) x = –1; y = 3; z = 0 C) x = 0; y = −1; z = 3 D) x = 0; y = 3; z = −1 E) x = –1; y = 2; z = 1

To solve this problem you must apply the proccedure shown below:

1. You have the following expression given in the problem above:

[tex] mn - 4m - 5n + 20 [/tex]

2. The first thing you must do to factor the expression is to group the terms, as following:

[tex] (mn-4m)+(-5n+20) [/tex]

3. Now, choose the greatest common factor of each group:

[tex] m(n-4)-5(n-4) [/tex]

4. Now, you need to factor the expression by factoring out the [tex] n-4 [/tex]

[tex] (m-5)(n-4) [/tex]

The answer is: [tex] (m-5)(n-4) [/tex]

Answer:

B) 46

Step-by-step explanation:

What you have to do here is plug in the numbers 2, 3, 4, and 5 in for i in the equation and then add the sums together:

3(2) + 1 = 7

3(3) + 1 = 10

3(4) + 1 = 13

3(5) + 1 = 16

Add those together and they equal 46

There is a vertical asymptote at x = 0 (because the denominator becomes 0). We need to further analyze the behavior at x = 3 to see if it's a hole (cancellation) or the function is continuous there (Option A or B might be the answer).

Step 1: Identify Potential Discontinuities

A rational function (like f(x)) is discontinuous when its denominator is zero. So, we need to find the values of x that make the denominator 0:

Denominator = 4x^2 - 12x = 0

Factor the denominator: 4x(x - 3) = 0

This gives us two possible values for x: x = 0 and x = 3.

Step 2: Verify Discontinuity Type at Each Point

Check if the function is defined (has a value) at these points.

x = 0: If we plug x = 0 in the original function, we get 0/0, which is indeterminate. This suggests a potential vertical asymptote at x = 0.

x = 3: If we plug x = 3 in the original function, the expression doesn't become 0/0. The function might have a hole or be continuous at x = 3.

Step 3: Analyze for Holes (Optional):

To further investigate x = 3, we can try to simplify the function by factoring the numerator (possible because the denominator is already factored). If the simplification gets rid of the term that causes the 0/0 at x = 3, it creates a hole.

They can be arranged in 210 ways.

We have given that, 3 identical emeralds 2 identical diamonds and 2 different opals

The arrangement of n objects where n₁ is one kind, n₂ is a second kind, and n₃ is a third kind is given by

[tex]\frac{n!}{n_1!*n_2!*n_3!}[/tex]

The total number of gems =7

3 of the first kind, 2 of the second kind and 2 of the third kind

Therefore, we get

[tex]\frac{7!}{2!*2!*3!}=210[/tex]

Therefore ,they can be arranged in 210 ways.

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Answer: B. n = 6 2/3

Hope this helps!

Answer with Step-by-step explanation:

We are given that:

f(x)=ax²+bx+c

also at x=0 f(x)=15

Putting x=0 in f(x)

a×0+b×0+c=15

⇒ c=15

now, f(2)=15.5

⇒ a×2²+b×2+c=15.5

⇒ 4a+2b+15=15.5

⇒ 4a+2b=0.5  -------(1)

f(4)=17

⇒ a×4²+b×4+c=17

⇒ 16a+4b+15=17

⇒ 16a+4b=2     --------(2)

equation (2)- 2×equation(1)

⇒ 16a+4b-8a-4b=2-2×0.5

⇒ 8a=2-1

⇒ 8a=1

⇒ a=1/8

⇒ a=0.125

Putting value of a and c in equation (1)

⇒ 4×0.125+2b=0.5

⇒ 0.5+2b=0.5

⇒ 2b=0

⇒ b=0

Hence

a=0.125

b=0

c=15

and f(x)=0.125x²+15

[tex]\text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a} [/tex]

a = 4

b = - 131

c = 480

The solutions of the trigonometric equation are approximately 0.56, 2.58, 3.71 and 5.72, all in concordance with the results found beside the analytical approach.

Trigonometric functions are periodic functions, cosine function has a standard period of 2π radians. There are two approaches to solve the expression given, an analytical one and a graphical one. Now we proceed to show each approach:

Analytical method

Now we proceed to apply algebraic handling and inverse trigonometric functions to solve the expression for t:

7 · cos 2t = 3

cos 2t = 3/7

2 · t = cos⁻¹ (3/7)

There are two solutions:

2 · t₁ ≈ 0.359π + 2π · i  ∨   2 · t₂ ≈ 1.641π + 2π · i

t₁ ≈ 0.180π + π · i   ∨ t₂ ≈ 0.821π + π · i

t₁₁ ≈ 0.180π ∨ t₁₂ ≈ 1.180π ∨ t₂₁ ≈ 0.821π ∨ t₂₂ ≈ 1.821π

The solutions of the trigonometric equations in [0, 2π] are approximately 0.18π, 0.82π, 1.18π and 1.82π.

Graphical approach

First, we add the definition of the following two functions (f(x) = 7 · cos 2t and g(x) = 3) and plot in the calculator through a graphing tool. The solutions of this trigonometric function are the points of intersection of the two functions.

According to the image attached below, the solutions of the trigonometric equation are approximately 0.56, 2.58, 3.71 and 5.72, all in concordance with the results found beside the analytical approach.

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Answer:

Without exponents, we can write the term as:  [tex]27(n)(n)(n)[/tex]

Step-by-step explanation:

The given term is :

[tex](3n)^{3}[/tex]

This can be written as:

[tex](3n)(3n)(3n)[/tex]

= [tex]27n^{3}[/tex]

= [tex]27(n)(n)(n)[/tex]

Hence, without exponents, we can write the term as:  [tex]27(n)(n)(n)[/tex]

Final answer:

The range of the function y = -x^2 + 1 is all real numbers less than or equal to 1, represented as ]-∞, 1].

Explanation:

The range of the function y = -x^2 + 1 refers to the set of all possible values that y can take as x varies over all real numbers. Since the function is a downward-opening parabola (due to the negative coefficient of x^2), the maximum value of y is at the vertex of the parabola. The vertex, in this case, occurs at x=0, which gives us the maximum y value of y = 1. As x moves away from zero in either direction, the value of y decreases and approaches negative infinity. Hence, the range of this function is ]-∞, 1], or all real numbers less than or equal to 1.

Answer:

70

Step-by-step explanation:

four books can be made from eight different books

4 books to be chosen from 8 books

The order does not matters so we use Combination.

four books can be made from eight different books, so we find 8C4

[tex]nCr= \frac{n!}{r!(n-r)!}[/tex]

[tex]8C4= \frac{8!}{4!(8-4)!}[/tex]

[tex]8C4= \frac{8!}{4!(4)!}[/tex]

[tex]8C4= \frac{8*7*6*5*4!}{4!(4)!}=70[/tex]

Answer:

70

Step-by-step explanation:

The number of BTUs does 1 cubic foot of propane yield is 684,000 BTUs/ft³ (in thousand)

Since we know that

The volume of the spherical tank is

[tex]V=4\div 3\pi r^3[/tex]

Now the volume of the tank will be:

[tex]V=4\div 3\times \pi \times (10/2)^3[/tex]

=523.59877756

= 523.6 ft³

Now the number should be

= [total amount] ÷ [total volume]

=358,000,000 ÷ 523.6

=683,728.0367 BTUs/ft³

= 684,000 BTUs/ft³ (in thousand)

Therefore, The number of BTUs does 1 cubic foot of propane yield is 684,000 BTUs/ft³ (in thousand)

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