Mathematics, 28.06.2019 01:00 gmanbom

Ms. smith spent 7 hours driving from her home to albany, a distance of 238 mi. before noon, she averaged 32 mph. after noon she averaged 39 mph. find the number of hours she traveled at each rate of speed.

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Answer from: mraymundo025p0gpw9

ANSWER

She averaged at 32 mph for 5 hours and 39 mph for 2 hours.

EXPLANATION

Let

be the number of hours she travelled before noon and

be the number of hours she travelled after noon.

Then since she travelled for a total of 7 hours, we can write the equation,

x + y = 7.....eqn(1)

We were also given that, before noon,she averaged at 32 mph.

We know that,

$speed = \frac{distance}{time \: taken}$

Let
d_1

be the distance before noon.

$\Rightarrow \: 32 = \frac{d_1}{x}$
This implies that,

d_1 = 32x

Also let the distance she covered after noon be,

d_2

Then,

$39 = \frac{d_2}{y}$
This implies that,

d_2 = 39y

Since she covered a total distance of 238 miles, we can write the equation,
d_1 + d_2 = 238

This means that,

32x + 39y = 238....eqn(2)

We multiply equation by 32 to get,

32x + 32y = 224...eqn(3)

Equation (2) minus equation (3), will give us,

7y = 14

This implies that,
y = 2

We substitute the value of y into equation (1) to get,

x + 2 = 7

We can now conclude that she averaged at 32 mph for 5 hours and 39 mph for 2 hours.

Answer from: Quest

x= -3 and $x=\frac{-3+/-3i\sqrt{3} }{2}$

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

$(x^3-a^3)=$ $(x-a)(x^2+ax+a^2)$

where a is the cube root of $a^3$ .

here this is 27 and the cube root of 27 is 3. so we write using the form:

$(x-3)(x^2+(3)x+3^2)\\(x-3)(x^2+3x+9)$ .

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

$x=\frac{-b+/-\sqrt{b^2-4ac} }{2a}$

where a = 1, b=3, and c=9

$x=\frac{-3+/-\sqrt{3^2-4(1)(9)} }{2(1)} \\x=\frac{-3+/-\sqrt{9-36)} }{2} \\x=\frac{-3+/-\sqrt{-27)} }{2}\\x=\frac{-3+/-3i\sqrt{3} }{2}$

this cubic has two complex roots.

Answer from: Quest

solution:

the given function is ,

$f(x) = 4 x^3 - 72 x^2 + 320 x,$

= $4x( x^2- 18 x + 80)\\\\ 4 x(x^2-10 x -8 x + 80) \\\\ 4 x[(x-10)(x-8)]$

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

$f(x) = a x^3+ b x^2 + c x+d,$

sum of the roots = a + b +c= 0+8+10=18= $\frac{-b}{a}$

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 = $\frac{c}{a}$

product of roots = a b c = 0= $\frac{d}{a}$

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function $f(x) = 4 x^3 - 72 x^2 + 320 x,$ is given by entry and exit route of this hall.