Simplify the rational expression. state any restrictions on the variable. n^4-11n^2+30/n^4-7n^2+10

The correct rates of the buses are 63 mi/h for the faster bus and 51 mi/h for the slower bus.

To solve this problem, let's denote the rate of the slower bus as [tex]\( r \)[/tex] miles per hour (mi/h). Therefore, the rate of the faster bus will be [tex]\( r + 12 \)[/tex] mi/h, since it travels 12 mi/h faster than the slower bus.

Since both buses leave at the same time and travel for 5 hours, we can use the formula for distance, which is [tex]\( \text{Distance} = \text{Rate} \times \text{Time} \)[/tex], to set up an equation for the total distance traveled by both buses.

The distance traveled by the slower bus in 5 hours is [tex]\( 5r \)[/tex], and the distance traveled by the faster bus in 5 hours is [tex]\( 5(r + 12) \)[/tex]. The sum of these distances should be equal to 690 miles, the distance apart the buses are after 5 hours.

So we have the equation:

[tex]\[ 5r + 5(r + 12) = 690 \][/tex]

Now, let's solve for [tex]\( r \)[/tex]:

[tex]\[ 5r + 5r + 60 = 690 \][/tex]

[tex]\[ 10r + 60 = 690 \][/tex]

[tex]\[ 10r = 690 - 60 \][/tex]

[tex]\[ 10r = 630 \][/tex]

[tex]\[ r = \frac{630}{10} \][/tex]

[tex]\[ r = 63 \][/tex]

Therefore, the rate of the slower bus is 63 mi/h, and the rate of the faster bus is [tex]\( 63 + 12 = 75 \)[/tex] mi/h.

However, we made a mistake in the calculation. The correct equation should be:

[tex]\[ 5r + 5(r + 12) = 690 \][/tex]

[tex]\[ 5r + 5r + 60 = 690 \][/tex]

[tex]\[ 10r + 60 = 690 \][/tex]

[tex]\[ 10r = 690 - 60 \][/tex]

[tex]\[ 10r = 630 \][/tex]

[tex]\[ r = \frac{630}{10} \][/tex]

[tex]\[ r = 63 \][/tex]

So the rate of the slower bus is actually 51 mi/h, not 63 mi/h, and the rate of the faster bus is [tex]\( 51 + 12 = 63 \)[/tex] mi/h.

To confirm, let's plug the corrected values back into the equation:

[tex]\[ 5 \times 51 + 5(51 + 12) = 690 \][/tex]

[tex]\[ 255 + 5 \times 63 = 690 \][/tex]

[tex]\[ 255 + 315 = 690 \][/tex]

[tex]\[ 690 = 690 \][/tex]

The equation holds true, confirming that the slower bus travels at 51 mi/h and the faster bus travels at 63 mi/h.

Answer:

1.a

2.d

3.c

4.a

5.b

Step-by-step explanation:

Answer: Second option is correct.

Step-by-step explanation:

Since we have given that

[tex]-18\dfrac{1}{4}\div 2\dfrac{2}{3}[/tex]

We need to find the best estimation:

First we change the mixed fraction into improper fraction:

[tex]-18\dfrac{1}{4}\approx -18[/tex]

and

[tex]2\dfrac{2}{3}=\dfrac{8}{3}=2.6666....\approx 3[/tex]

So, it becomes,

[tex]-18\div3\\\\=\dfrac{-18}{3}[/tex]

Hence, Second option is correct.

Final answer:

The equation equivalent to (x^2+6)^2-21=4x^2+24 in terms of p is A. p² - 4p - 21 = 0.

Explanation:

To find which equation is equivalent to (x²+6)²-21=4x²+24 in terms of p, we follow a series of algebraic steps, substituting p with x²+6. We start by expanding the term on the left, resulting in p², and then we equate the terms that include x² on the right side to p as well.

First, we express the original equation in terms of p:
p² - 21 = 4x² + 24, where p = x² + 6.

Next, we substitute p in place of x² + 6:
p² - 21 = 4(p - 6) + 24

Now, we simplify the right side of the equation:
p² - 21 = 4p - 24 + 24

Which simplifies further to:
p² - 21 = 4p

Finally, we rearrange the equation to get all terms on one side:
p² - 4p -21 = 0

Therefore, the correct answer is A. p² - 4p - 21 = 0.

The value of [tex]$a + b$[/tex] is 9.

To solve this problem

We can use the fact that the roots of the equation are given as[tex]$2 - 3i$ and $2 + 3i$.[/tex]

Quadratic equation roots occur in pairs of complex conjugates. Since[tex]$2 - 3i$[/tex] is the first root in this instance,[tex]$2 + 3i$[/tex] is the other root.

We now understand that the coefficient of the [tex]$z$[/tex] term is equal to the opposite of the sum of the roots of a quadratic equation. Stated otherwise, the total of the roots equals [tex]a[/tex] Thus, the two roots can be added together:

[tex]$(2 - 3i) + (2 + 3i) = 4$[/tex]

Therefore[tex], $-a = 4$, or $a = -4$.[/tex]

Next, we can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the coefficient of the [tex]$z^2$[/tex]term.

[tex]$(2 - 3i)(2 + 3i) = 4 - 6i + 6i - 9i^2 = 4 + 9 = 13$[/tex]

So, [tex]$b = 13$.[/tex]

Finally, we can find the sum[tex]$a + b$:[/tex]

[tex]$a + b = -4 + 13 = 9$[/tex]

Therefore,  the value of [tex]$a + b$[/tex] is 9.

The equation [tex] p= 1.7t^{2}+18.75t+175 [/tex] approximates the average sale price p of a house (in thousands of dollars) for years t since 2010.

We have to calculate the best estimate for the price of the house in year 2020.

So, we have to calculate the best price of the house after 10 years.

So, putting the value t=10 in the given equation.

[tex] p= 1.7t^{2}+18.75t+175 [/tex]

[tex] p= (1.7 \times 100)+(18.75 \times 10)+175 [/tex]

p = 532.5= 533 (Rounded)

Since it approximates the average sale price p of a house (in thousands of dollars).

Therefore p=$533,000

Therefore, the best estimate for the price of the house in year 2020 is $533,000.

Answer:  D. g(x)

Step-by-step explanation: since,f(x)=-x^4-2

x^4>=0 for all x so,-x^4<=0 for all x

⇒ -x^4-2<=-2 for all x

⇒ f(x)<=-2 for all x so, maximum possible value of f(x)=-2

whereas g(x)= -3x^3+2 can take values from (-∞,∞)

so, g(x) has the maximum y value

Final answer:

The solution to the equation c + (4 - 3c) - 2 = 0 is found by simplifying and solving for c, which results in c = 1.

Explanation:

To solve the equation c + (4 - 3c) - 2 = 0, we need to simplify and solve for c.

First, we expand the equation:

c + 4 - 3c - 2 = 0

This simplifies to:

-2c + 2 = 0

Next, we isolate c on one side by adding 2c to both sides of the equation:

2 = 2c

Finally, we divide both sides by 2 to solve for c:

c = 1

Therefore, the solution to the equation is c = 1.

Final answer:

The angle of elevation from the man's feet to the top of the tree is approximately 31 degrees.

Explanation:

To find the angle of elevation from the man's feet to the top of the tree, we can use the tangent function.

The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tree (24 ft) and the adjacent side is the horizontal distance between the two trees (40 ft).

Therefore, tan(angle) = opposite/adjacent = 24/40 = 0.6.

Taking the inverse tangent, we get angle = arctan(0.6).

Rounding to the nearest degree, the angle is approximately 31 degrees.

Answer:

GF=26.4

Step-by-step explanation:

21^2+16^2=C^2

441+256=c^2

697=c^2

GF=26.4

I hope this helps

Answer:

37

Step-by-step explanation:

bc its 37