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2 years ago

13 Answer:

Mathematics

1 answer:

Marizza181 [45]2 years ago

4 0

Step-by-step explanation:

13.

132 miles in 3 h = 132miles/3hours = 132/3 miles/h =

= 44 miles/h

so,

how long for 110 miles ?

speed = distance/time

time = distance/speed = 110 miles / 44 miles/h =

= 110/44 h = 55/22 h =

= 5/2 h = 2.5 h

14.

60% = 60/100 = 6/10 = 3/5

15.

300% = 300/100 = 3

16.

116 2/3% = (348 + 2)/3% = 350/3% = 350/3/100 =

= 350/300 = 1 50/300 = 1 1/6

17.

19% = 19/100

18.

3.8% = 3.8/100 = 38/1000 = 19/500

19.

166 2/3% = (498 + 2)/3% = 500/3% = 500/3/100 =

= 500/300 = 5/3 = 1 2/3

20.

4/5 = 0.8 = 80%

21.

7/4 = 1.75 = 175%

22.

1/3 = 0.3333333... ≈ 33.33%

23.

2 = 200/100 = 200%

24.

0.4 = 40%

25.

0.375 = 37.5/100 = 37.5%

26.

80% of 60 = 60×80/100 = 60×4/5 = 48

27.

24% of 65 = 65×24/100 = 65×6/25 = 13×6/5 =

= 15.6

28.

115% of 138 = 138×115/100 = 138×23/20 =

= 69×23/10 = 158.7

29.

18.3% of 74 = 74×18.3/100 = 74×183/1000 =

= 37×183/500 = 13.542 ≈ 13.54

30.

6.5% of 115 = 115×6.5/100 = 115×65/1000 =

= 23×13/40 = 7.475 ≈ 7.48

31.

0.75% of 93 = 93×0.75/100 = 93×75/10000 =

= 93×3/400 = 0.6975 ≈ 0.70

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Let f be the function defined by f(x)=cx−5x^2/2x^2+ax+b, where a, b, and c are constants. The graph of f has a vertical asymptot

Musya8 [376]

Answer:

a) $a = 2$ and $b = -4$ , b) $c = -10$ , c) $f(-2) = -\frac{5}{3}$ , d) $y = -\frac{5}{2}$ .

Step-by-step explanation:

a) After we read the statement carefully, we find that rational-polyomic function has the following characteristics:

1) A root of the polynomial at numerator is -2. (Removable discontinuity)

2) Roots of the polynomial at denominator are 1 and -2, respectively. (Vertical asymptote and removable discontinuity.

We analyze each polynomial by factorization and direct comparison to determine the values of $a$ , $b$ and $c$ .

Denominator

i) $(x+2)\cdot (x-1) = 0$ Given

ii) $x^{2} + x-2 = 0$ Factorization

iii) $2\cdot x^{2}+2\cdot x -4 = 0$ Compatibility with multiplication/Cancellative Property/Result

After a quick comparison, we conclude that $a = 2$ and $b = -4$

b) The numerator is analyzed by applying the same approached of the previous item:

Numerator

i) $c\cdot x - 5\cdot x^{2} = 0$ Given

ii) $x \cdot (c-5\cdot x) = 0$ Distributive Property

iii) $(-5\cdot x)\cdot \left(x-\frac{c}{5}\right)=0$ Distributive and Associative Properties/ $(-a)\cdot b = -a\cdot b$ /Result

As we know, this polynomial has $x = -2$ as one of its roots and therefore, the following identity must be met:

i) $\left(x -\frac{c}{5}\right) = (x+2)$ Given

ii) $\frac{c}{5} = -2$ Compatibility with addition/Modulative property/Existence of additive inverse.

iii) $c = -10$ Definition of division/Existence of multiplicative inverse/Compatibility with multiplication/Modulative property/Result

The value of $c$ is -10.

c) We can rewrite the rational function as:

$f(x) = \frac{(-5\cdot x)\cdot \left(x+2 \right)}{2\cdot (x+2)\cdot (x-1)}$

After eliminating the removable discontinuity, the function becomes:

$f(x) = -\frac{5}{2}\cdot \left(\frac{x}{x-1}\right)$

At $x = -2$ , we find that $f(-2)$ is:

$f(-2) = -\frac{5}{2}\cdot \left[\frac{(-2)}{(-2)-1} \right]$

$f(-2) = -\frac{5}{3}$

d) The value of the horizontal asympote is equal to the limit of the rational function tending toward $\pm \infty$ . That is:

$y = \lim_{x \to \pm\infty} \frac{-10\cdot x-5\cdot x^{2}}{2\cdot x^{2}+2\cdot x -4}$ Given

$y = \lim_{x \to \infty} \left[\left(\frac{-10\cdot x-5\cdot x^{2}}{2\cdot x^{2}+2\cdot x-4}\right)\cdot 1\right]$ Modulative Property

$y = \lim_{x \to \infty} \left[\left(\frac{-10\cdot x-5\cdot x^{2}}{2\cdot x^{2}+2\cdot x-4}\right)\cdot \left(\frac{x^{2}}{x^{2}} \right)\right]$ Existence of Multiplicative Inverse/Definition of Division

$y = \lim_{x \to \pm \infty} \left(\frac{\frac{-10\cdot x-5\cdot x^{2}}{x^{2}} }{\frac{2\cdot x^{2}+2\cdot x -4}{x^{2}} } \right)$ $\frac{\frac{x}{y} }{\frac{w}{z} } = \frac{x\cdot z}{y\cdot w}$

$y = \lim_{x \to \pm \infty} \left(\frac{-\frac{10}{x}-5 }{2+\frac{2}{x}-\frac{4}{x^{2}} } \right)$ $\frac{x}{y} + \frac{z}{y} = \frac{x+z}{y}$ / $x^{m}\cdot x^{n} = x^{m+n}$

$y = -\frac{5}{2}$ Limit properties/ $\lim_{x \to \pm \infty} \frac{1}{x^{n}} = 0$ , for $n \geq 1$

The horizontal asymptote to the graph of f is $y = -\frac{5}{2}$ .

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