A research submarine dives at a speed of 100 ft/min directly toward the research lab. How

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Firlakuza [10]

Answer:

Domain: All the real values except x = 2 and x = 4: R - {2, 4}
Holes: x = 2
VA, vertical asymptores: x = 4
HA: horizontal asymptotes: there are not horizontal asymptotes
OA: oblique asymptotes: x + 6 [note that OH does not stand for any known feature, and so it is understood that it was intended to write OA]
Roots: x = 2
Y-intercept: -1

Step-by-step explanation:

1. Given:

$f(x)=\frac{x^3-8}{x^2-6x+8}$

Note that the number 8 in the numerator is not part of the power.

Type of function: rational function

2. Domain: is the set of x-values for which the function is defined.

The given function is defined for all x except those for which the denominator equals 0.

Denominator: x² -6x + 8 = 0

Solve for x:

Factor. (x - 4 )(x - 2) = 0

Zero product property: (x - 4) = 0 or (x - 2) = 0

x - 4 = 0 ⇒ x = 4

x - 2 = 0 ⇒ x = 2

Domain:

All the real values except x = 2 and x = 4: x ∈ R / x ≠ 2 and x ≠ 4.

3. Holes:

The holes on the graph of a rational function are at those x-values for which both the numerator and denominator are zero.

Find the values for which the numerator is zero:

Numerator: x³ - 8 = 0

Factor using difference of cubes property:

a³ - b³ = (a - b)(a² + ab + b²)

x³ - 8 = (x - 2)(x² + 2x + 4) = 0

Zero product property: (x - 2)(x² + 2x + 4) = 0

x - 2 = 0 ⇒ x = 2

x² + 2x + 4 = 0 (this has not real solution)

The values for which the denominator is zero were determined above: x = 2 and x = 4.

Conclusion: for x = 2 both numerator and denominator equal 0, so this is a hole.

4. VA: Vertical asymptotes.

The vertical asymptotes on the graph of a rational function are the vertical lines for which only the denominator (and not the numerator) equals zero.

In the previous part it was determined that happens when x = 4.

5. HA: Horizontal asymptotes.

In rational functions, if the numerator is a higher degree polynomial than the denominator, there is no horizontal asymptote.

6. OA: oblique asymptotes

Find the quotient and the remainder.

x + 6

_______________

x² - 6x + 8 ) x³ + 0x² + 0x - 8

- x³ + 6x² - 8x

___________

6 x² - 8x - 8

- 6x² + 36x - 48

_____________

28x - 56

Result: (x + 6) + (28x - 56) / (x² - 6x + 8)

Find limit x → ∞

$\lim_{x \to \infty}(x + 6) + \frac{28x-56}{x^2-6x+8}=x+6$

7. Roots:

Roots are the values for which f(x) = 0.

That happens when the numerator equals 0, and the denominator is not 0.

As determined earlier: x³ - 8 = 0 ⇒ x = 2.

8. Y-Intercept

The y-intercepts of any function are the y-values when x = 0

Substitute x = 0 into the function:

$f(x)=\frac{x^3-8}{x^2-6x+8}=\frac{0^3-8}{0^2-6(0)+8}}=\frac{-8}{8} =-1$

3 0

3 years ago

Anna has a 30-year, 5.75% mortgage on her $250,000 home. She has been

poizon [28]

Answer:

Her new monthly payment is now $1,378.91¢

Step-by-step explanation:

For us to calculate the new monthly mortgage payment that Anna will start paying from now on, we need to input the formula for calculating monthly mortgage payments.

The formula is:-

$M = P [\frac{r(1+r)^{n} }{(1+r)^{n}-1}]$

Where M is the monthly mortgage payment.

P is the principal

r is the monthly interest rate calculated by dividing your annual interest rate by 12

n is the number of payments(the number of months you will be paying the loan).

In this case, the new principal that Anna must pay back is $231,905.47¢. The annual interest rate has been reduced to 5.17% from 5.75% so the new monthly interest rate will be obtained by dividing the new annual interest rate by 12

= 5.17%/2

= 0.431%

This is the new monthly interest rate.

Since she has been paying her mortgage loan diligently for 5 complete years. It means she now has just 25 years to complete the payment. If 12 months make up one year, then there are - 12 × 25 = 300 more months to go.

300 is therefore "n" that is required for the calculation.

All the terms needed for the calculation of her new monthly mortgage is now complete.

P = $231,905.47¢

r = 0.431%

n = 300

$M = 231,905.47[\frac{0.00431(1+0.00431)^{300} }{(1+0.00431)^{300} -1}]$

$= 231,905.47[\frac{0.00431(3.634)}{2.634}]$

= 231,905.47 × 0.005946

M = $1,378.91¢

Therefore her new monthly mortgage payment will become $1,378.91¢

4 0

3 years ago

Read 2 more answers

The surface area of a triangular pyramid is 80 m2. If the pyramid is enlarged by a scale factor of 2, what is the surface area o

astra-53 [7]

160 m^2 is the answer

4 0

3 years ago

Read 2 more answers

bag contains 15 marbles numbered from 1 to 15. Two marbles are selected from the bag at the same time. Let event E = the two num

lbvjy [14]

The numbers DO differ by 10 if they are

- 1 and 11
- 2 and 12
- 3 and 13
- 4 and 14
- 5 and 15

Each outcome technically has two ways of occurring, but since we're taking two marbles at a time, that would be the same as saying, for instance, that drawing 1 and 11 is the same as drawing 11 and 1. We only count such an event once.

So the answer is (a) 5.

7 0

3 years ago

Read 2 more answers

The larger of two numbers is 1 more than twice the

stellarik [79]

Answer:

Step-by-step explanation:

Hi there!

Let the smaller number be x and the larger number be 2x+1.

According to the question,

The sum of the numbers is 20 less than three

times the larger.

i.e x + (2x+1)= 3*(2x+1) - 20

or, x + 2x +1 = 6x + 3 -20

or, 3x + 1 = 6x - 17

or, 3x = 18

or x= 18/3

x = 6

And 2x+1 = 2*6+1 = 13.

Therefore the smaller number is 6 and larger number is 13.

Hope it helps!

7 0

2 years ago