A candy company used 8 gallons of syrup to make 4 batches of candy. what is the rate of syrup per batch.

Answer:

The equation has one real root and two complex roots.

Step-by-step explanation:

The given equation is

[tex]x^3-3x^2+4x-12=0[/tex]

The above equation is true form x=3, therefore (x-3) is a factor of above equation.

Use long division or synthetic division method to divide the equation by (x-3).

[tex](x-3)(x^2+4)=0[/tex]

Equate each factor equal to zero.

[tex]x-3=0[/tex]

[tex]x=3[/tex]

Therefore 3 is a real root of the equation.

[tex]x^2+4=0[/tex]

[tex]x^2=-4[/tex]

[tex]x=\sqrt{-4}[/tex]

[tex]x=\pm 2i[/tex]

2i and -2i are complex roots of the equation.

Therefore the equation has one real root and two complex roots.

The valid postulates and theorems for proving triangles congruent are ASA, AAS, and SAS. The SSA postulate is not a valid postulate.

The valid postulates and theorems for proving triangles congruent are ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and SAS (Side-Angle-Side).

The SSA (Side-Side-Angle) postulate, also known as the ambiguous case of the Law of Sines, is not a valid postulate for proving triangles congruent because it can result in two different triangles or no triangle at all.

Thus, the correct answer is A. SSA.

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The value of AC + BD will be 70.

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

Given that;

In rectangle ABCD,

AB = 21

AD = 28

Now,

Since, ABCD is a rectangle.

Hence, We get;

⇒ AC = BD

Here, In rectangle ABCD,

AB = 21

AD = 28

So, By using definition of Pythagoras theorem, we get;

⇒ BD² = AB² + AD²

⇒ BD² = 21² + 28²

⇒ BD² = 441 + 784

⇒ BD² = 1225

⇒ BD = 35

Thus, The value of BD + AC is find as;

⇒ AC + BD = 35 + 35

                 = 70

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Answer: 12 < x < 15

Step-by-step explanation:

cos4x = cos2x

We know that:

cos2x = 1-2cos^2 x

==> cos4x = 1-2cos^2 (2x)

Now substitute:

==> 1-2cos^2 (2x) = cos2x

==> 2cos^2 (2x) + cos2x - 1 = 0

Now factor:

==> (2cos2x -1)(cos2x + 1) = 0

==> 2cos2x -1 = 0 ==> cos2x =1/2 ==> 2x= pi/3

==> x1= pi/6 , 7pi/6

==> x1= pi/6 + 2npi

==> x2= 7pi/6 + 2npi

==> cos2x = -1 ==> 2x= pi ==> x3 = pi/2 + 2npi.

==> x= { pi/6+2npi, 7pi/6+2npi, pi/2+2npi}

The solutions to the equation cos(4x) + cos(2x) - 2 = 0 are x = nπ and x = (2n + 1)π/2, where n is an integer.

Here, we have to solve the equation cos(4x) + cos(2x) - 2 = 0, we can use trigonometric identities to simplify it.

We know the following trigonometric identities:

[tex]cos(2x) = 2cos^2(x) - 1\\cos(4x) = 2cos^2(2x) - 1[/tex]

Now, let's rewrite the equation using these identities:

[tex]2cos^2(2x) - 1 + 2cos^2(x) - 1 - 2 = 0[/tex]

Combine like terms:

[tex]2cos^2(2x) + 2cos^2(x) - 4 = 0[/tex]

Divide the entire equation by 2:

[tex]cos^2(2x) + cos^2(x) - 2 = 0[/tex]

Now, let's simplify further:

We can rewrite [tex]cos^2(2x)[/tex] as [tex](1 - sin^2(2x))[/tex]using the Pythagorean identity: [tex]sin^2(\theta) + cos^2(\theta) = 1[/tex]

So, the equation becomes:

[tex](1 - sin^2(2x)) + cos^2(x) - 2 = 0[/tex]

Rearrange the terms:

[tex]cos^2(x) - sin^2(2x) - 1 = 0[/tex]

Now, use the double-angle identity: [tex]sin^2(2x) = 2sin(x)cos(x)[/tex]

[tex]cos^2(x) - 2sin(x)cos(x) - 1 = 0[/tex]

Now, we can rewrite [tex]cos^2(x) as (1 - sin^2(x))[/tex] using the Pythagorean identity:

[tex]sin^2(\theta) + cos^2(\theta) = 1[/tex]

[tex](1 - sin^2(x)) - 2sin(x)cos(x) - 1 = 0[/tex]

Now, simplify further:

[tex]1 - sin^2(x) - 2sin(x)cos(x) - 1 = 0[/tex]

We can see that [tex](1 - sin^2(x))[/tex] cancels out:

-2sin(x)cos(x) = 0

Divide both sides by -2:

sin(x)cos(x) = 0

Now, there are two possible solutions for this equation:

sin(x) = 0

cos(x) = 0

Let's solve each case:

sin(x) = 0

This occurs when x is an integer multiple of π (π, 2π, 3π, ...). So the solutions are x = nπ, where n is an integer.

cos(x) = 0

This occurs when x is an odd multiple of π/2 (π/2, 3π/2, 5π/2, ...). So the solutions are x = (2n + 1)π/2, where n is an integer.

Therefore, the solutions to the equation cos(4x) + cos(2x) - 2 = 0 are x = nπ and x = (2n + 1)π/2, where n is an integer.

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Answer:

It would take 2 seconds to the key to reach the ground.

The domain the function is: 0≤t≤2 or in interval notation [0,2].

The range of the function is: 0≤h≤64 or in interval notation [0,64].

Step-by-step explanation:

The provided function is: [tex]h=-16t^2 +64[/tex]

Where h represents the height and t represents the time.

Part(A) how long does it take the key to reach the ground?

When key will be at ground, the height will be 0.

Substitute the value of h=0 in above equation and solve for t.

[tex]-16t^2 +64=0[/tex]

[tex]-16(t^2 -4)=0[/tex]

[tex](t^2 -2^2)=0[/tex]

[tex](t+2)(t-2)=0[/tex]

[tex]t+2=0\ or\ t-2=0[/tex]

[tex]t=-2\ or\ t=2[/tex]

Ignore the negative value of t.

Hence, it would take 2 seconds to the key to reach the ground.

Part (B) what are the reasonable domain and range for the function h?

Time and Distance should be a positive number.

Thus, the value of t and h should be greater or equal to 0.

From the part (a) we know that the after 2 second the key will hit the ground so the value of t can be greater or equal to 0 but should be less or equal to 2.

Thus, the domain the function is: 0≤t≤2 or in interval notation [0,2].

The maximum height of the key will be at t=0.

Thus the maximum value of h can be 64 and minimum value can be 0.

The range of the function is: 0≤h≤64 or in interval notation [0,64].

Answer:

1. Which equation can be used to find the time to complete the job when x workers are assigned to it?

A.) m(x) = 270/x

2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25

What is H(x) when x =2?

B.) 6.25

Step-by-step explanation:

1. We know when x=3 (workers), m(x)= 90 min (time is takes to mow the lawn at a large park) and we know m(x) varies inversely with x, this means when m(x) increases so x decreases. This is obtained with a division over x.

Now, we can replace to know what is the correct answer:

[tex]m(x)=\frac{y}{x} (1) \\90=\frac{y}{3} \\90*3=y[/tex]

[tex]y=270[/tex]

The answer is A.

[tex]m(x)=\frac{270}{x}[/tex]

We can confirm if we replace x=3 in the equation before. Our answer should be 90.

[tex]m(x)=\frac{270}{3}=90[/tex]

2.  Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25

We know H(x) varies inversely with x, this means when H(x) increases so x decreases. This is obtained with a division over x. Now, we need to find the constant in the numerator.

Now, we can replace in the eq (1) to know the constant in the numerator:

[tex]H(x)=\frac{y}{x}\\50=\frac{y}{0.25} \\50*0.25=y[/tex]

[tex]y=12.5[/tex]

Now, we can replace in the eq (1) to find H(x) when x=2

[tex]m(x)=\frac{y}{x}[/tex]

[tex]m(x)=\frac{12.5}{2}[/tex]

[tex]m(x)=6.25[/tex]  

The answer is B.) 6.25

The correct option is A: [tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]

To determine which option is equivalent to the given function f(x) = [tex]3(2)^{3x} \),[/tex] we will simplify the given function and compare it to each option.

The given function is:

[tex]\[ f(x) = 3(2)^{3x} \][/tex]

We can rewrite the exponent part using properties of exponents:

[tex]\[ (2)^{3x} = (2^3)^x = 8^x \][/tex]

So the function becomes:

[tex]\[ f(x) = 3 \cdot 8^x \][/tex]

Now, let's compare this to each of the given options:

Option A: [tex]\( f(x) = 3(8)^x \)[/tex]

This is exactly the same as our simplified function. Therefore, Option A is equivalent to [tex]\( f(x) = 3(2)^{3x} \).[/tex]

Option B: [tex]\( f(x) = 24^x \)[/tex]

This is not equivalent because it implies a different base raised to the power of x.

Option C:[tex]\( f(x) = 3(8x) \)[/tex]

This is not equivalent because it multiplies by 27 instead of 3.

Option D: [tex]\( f(x) = 3(8x) \)[/tex]

This is not equivalent because it implies multiplying 8 by x, not raising 8 to the power of x.

Therefore, the correct answer is:

[tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]

The complete question is here:

Which of the following options is an equivalent function to f(x) = 3(2)3x?

A. f(x) = 3(8)x

B. f(x) = 24x

C. f(x) = 27(8)x

D. f(x) = 3(8x)

Answer: The amount she will get is $4.10

Step-by-step explanation:

My reason is because .15 times 6 since she is buying 6 pencils, it is .90. The you subtract the decimal 5.00-0.90 and you get 4.10. Hope this helps. PLease mark brainliest!!!!!

Answer:

C.

Step-by-step explanation:

We are going to solve it with substitution, that is, we are going to clear one variable and then replace it in the other equation. Then,

[tex]3x + 2y = 12[/tex]

[tex]3x = 12-2y[/tex]

[tex]x = \frac{12-2y}{3}[/tex]

[tex]x = \frac{12}{3}-\frac{2y}{3}[/tex]

[tex]x = 4-\frac{2y}{3}.[/tex]

Now, we replace that x value in the equation 6x + 3y = 21:

[tex]6(4-\frac{2y}{3}) + 3y = 21[/tex]

[tex]24-\frac{12y}{3} + 3y = 21[/tex]

[tex]24-4y + 3y = 21[/tex]

[tex]24-y = 21[/tex]

[tex]-y = 21-24[/tex]

[tex]-y = -3[/tex]

[tex]y = 3.[/tex]

Finally, we replace the y value found to find x.

[tex]x = 4-\frac{2*3}{3}[/tex]

[tex]x = 4-2[/tex]

[tex]x = 2.[/tex]

So, the solution is (x,y)=(2,3). Then, the answer is C.

Answer:

The sensory somatic nervous system controls all voluntary responses of the body.

Step-by-step explanation:

The sensory somatic nervous system controls all voluntary responses of the body.

The movements includes muscles and organs movements and reflex movements. Here the sensory neurons carry impulses to the brain and the spinal cord.

Answer:

x = 3

Step-by-step explanation:

  Given:[tex]\log _{x-1}\left(16\right)=4[/tex]

We have to solve for x.

Consider the given expression [tex]\log _{x-1}\left(16\right)=4[/tex]

[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(b\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}[/tex]

[tex]\log _{x-1}\left(16\right)=\frac{\ln \left(16\right)}{\ln \left(x-1\right)}[/tex]

Thus, the expression becomes,

[tex]\frac{\ln \left(16\right)}{\ln \left(x-1\right)}=4[/tex]

Multiply both side by [tex]\ln \left(x-1\right)[/tex]

[tex]\frac{\ln \left(16\right)}{\ln \left(x-1\right)}\ln \left(x-1\right)=4\ln \left(x-1\right)[/tex]

Simplify, we get,

[tex]\ln \left(16\right)=4\ln \left(x-1\right)[/tex]

Divide both side by 4, we have,

[tex]\frac{4\ln \left(x-1\right)}{4}=\frac{\ln \left(16\right)}{4}[/tex]

[tex]\frac{\ln \left(16\right)}{4}:\quad \ln \left(2\right)[/tex]

Thus, [tex]\ln \left(x-1\right)=\ln(2)[/tex]

[tex]\mathrm{When\:the\:logs\:have\:the\:same\:base:\:\:}\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\quad \Rightarrow \quad f\left(x\right)=g\left(x\right)[/tex]

thus, x - 1 = 2

simplify for x, we have,

x = 3

Using the point slope equation, it is determined that Keisha will not finish reading the 325 pages book in 10 days if she is reading at a rate of 25 pages per day.

The subject of this question is Mathematics and it involves solving a problem using a point slope equation.

Let's denote the total number of pages in the book as 'y' and the number of pages that Keisha can read per day as 'm'. We know that y=325 and m=25. The point slope equation in this case would be y=mx, where 'x' represents the number of days. In this case, we are trying to find out whether x=10.

Substituting the known values into the equation we get: 325=25*10. This simplifies to 325=250.

Since 325 does not equal 250, we can conclude that Keisha will not complete reading the book in 10 days.

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