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**caymanisland****Member**- Registered: 2011-03-28
- Posts: 3

Brenda drives an 18 wheeler. she plans to buy her own truck. Her research indicates that the expected running costs, C, in dollars, per 100 km, are given by C(v)=0.9 +0.0016v square, where v is the speed, in kilometers per hour. Brenda's first trip will be 1500 km, round trip. she plans to pay herself $30/h. determine the speed that will minimize Brend's cost for the trip

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi caymanisland,

The cost of the trip is an increasing function, so I assume that you mean that the cost must not increase beyond 30 $/hr.

I solved the following equation, which has one real solution:

x is the speed in km/hr, so that cost is 30 $/hr. Any speed less than that would result in lower $/hr as well as lower cost of the trip.

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,858

Hi caymanisland;

Welcomes to the forum. Are you sure you copied the problem correctly? You should copy them word for word.

If we set up the equation of Brenda's cost and following gAr's idea.

v = 121.7901312369059 km/hr

But like him I am forced to make some assumptions. With this model there is no minimum.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

And it costs even when it's at rest.

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**Naru****Member**- Registered: 2012-03-24
- Posts: 1

gAr wrote:

Hi caymanisland,

The cost of the trip is an increasing function, so I assume that you mean that the cost must not increase beyond 30 $/hr.

I solved the following equation, which has one real solution:

x is the speed in km/hr, so that cost is 30 $/hr. Any speed less than that would result in lower $/hr as well as lower cost of the trip.

The answer in my textbook says it's 97.9km/h. Are you sure that's right?

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,525

hi caymanisland and Naru

Welcome, both of you, to the forum.

I'm interpreting this problem differently.

She has two costs when she makes trips.

(i) The running costs go up as the velocity of the truck increases.

(ii) But she is making a charge for her time (to the customers?) which goes up with the time.

So the overall cost goes higher with speed due to runnning costs but lower due to her time costs.

Anyone want to try that?

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,858

Hi Naru;

What is the name of the text book?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,525

hi bobbym,

Your post just got in ahead of mine. Maybe a different interpretation? Post 6

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,858

Hi Bob;

The only way I can see to get the answer they want is to treat her salary of $30 per hour as part of the cost. The question stating she pays herself suggests that.

We differentiate and set to 0.

Solving we get one real root of v =97.87 km / hr. That is the minimum velocity

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,525

hi caymanisland and Naru

Here's what I'm getting:

running cost (v) = 15(0.9 + 0.0016v^2) for this journey (15 lots of 100)

driver cost = (1500/v) x 30 (journey time x cost per hour)

so

so differentiate

Set this equal to zero and re-arrange

This is > 0 for the calculated v so this gives a minimum.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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