two word problems

Question

 

1)A car company charges $31 per day to rent a particular type of car and $0.16 per mile. Juan is charged $47.96 for a one day rental. How many mile did he drive?

a)121mi

b)106mi

c)300mi

d)269mi

 

2) The number if mosquitos M(x), in millions, in a certain area depends on the june rain fall x, in inches, according to the function M(x)=18.0x-x^2, what rainfall produces the maximum number of mosquitoes? round the answer to the nearest tenth.

a)9.0in

b)18.0in

c)324.0n

d)0.0in

 

 

Answer 1

For question 1, we translate english directly into math:  Cost = $31 + $0.16*miles or C = .16m + 31.  We are given the total cost C, so we can directly solve for m.

 

For question 2, the most direct way is to differentiate the equation with respect to x and set it equal to zero:

 

dM/dx = 18 -2x.  Setting equal to zero we get -2x+18 = 0 yielding x = 9 for the best amount of rainfall for mosquitoes.  We can be sure this is a true global maximum by the fact that the equation is a quadratic concave downward.

Answer 2

I'm not the greatest with math but the first one is simple enough!

 

47.96 (total charged)

31 (base rate for 1 day rental)

 

47.96 - 31 = 16.96

 

16.96 / .16 = 106 ; therefore, Juan drove 106 miles. The Answer is B.

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For the second question....

 

M(x) is mosquitos, in millions

 

x is rainfall in inches.

 

So from here, it's just plug-and-play:

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A) M(x) = 18.0 x (9.0) - 9.0^2 [squared]

M(x) = 162 - 81

 

First option is... M(x)=81

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B) M(x) = 18.0 x (18.0) - 18.0^2 [squared]

M(x) = 324 - 324 = 0

Second option is... M(x) = 0

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C) M(x) = 18.0 x (324.0) - 324.0^2 [squared]
M(x) = 5,832 - 104,976 = -99,144

Third option is... M(x)= -99,144

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D) M(x) = 18.0 x (0.0) - 0.0^2 [squared]
M(x) = 0 - 0 = 0

Last option is... M(x) = 0

 

If I've worked through this second equation correctly the answer to the second problem will be A

Answer 3

$47.96-$31.00=$16.96

$16.96/$0.16=1696/16=106 miles, b)

 

M(x)=18x-x^2

M(x)=x(18-x)

x>0 and x<18

0<x<18

x is between 0 and 18, not including 0 and not including 18

that eliminates b,c, and d

a) 9 in is between 0 and 18

Answer 4

1) 31+ 0.16m=47.96

.16m=16.96

 

m=106 miles b)

 

2) y=18x-x^2 is a parabola that opens down with a maximum at its vertex.

 

The x coordinate of the vertex is -b/2a where a b c are the coefficients of the polynomial y=ax^2+bx+c

In our case a=-1 b=18 c=0

so xv=-b/2a=-18/-2=9 

 

A rainfall of 9 inches produces the maximum.. which can be found out by plugging in 9 for x in the initial equation.

 

For extra help, look up youtube videos on finding maximums and minimums of quadratic functions/parabolas.