Question

QUADRATIC Word-Problem


Engineers are programming a robotic arm to move toward an object it will pick up. They define the way the arm will move by using the function
f(x) = -1/2x^2 - 2x + 16, where f(x) represents the distance between the arm and the object, and x represents the number of seconds that arm is in motion.

Determine the number of seconds it will take for the arm to reach the object it will pick-up.

Answer 1

@jhonyy9 @zepdrix

Answer 2

Does it mean for me to find the vertex?

Answer 3

f(x) represents distance
x represents time

we want to know the `time x` when the `distance between the objects is zero, f(x)=0`

Answer 4

\[\large\rm f(x) = -\frac12x^2 - 2x + 16\]\[\large\rm 0 = -\frac12x^2 - 2x + 16\]Solve for x, ya? :)

Answer 5

Okay!

Answer 6

Ooo this one factors nicely :o

Answer 7

I used the quadratic equation.

Answer 8

Fair enough :)

Answer 9

I got x = 8 , 4

Answer 10

Hmm you're missing a minus sign on one of those values...

Answer 11

sorry -8, 4

Answer 12

Niceeeee.
So which value(s) make sense?

At time -8 seconds, the arm has reached the object?

Answer 13

4

Answer 14

There's another part. It says: Describe how the engineers could modify the function so that it takes the arm one second longer to reach the object.

Answer 15

hmm

Answer 16

Well this is how I approached the problem,\[\large\rm 0 = -\frac12x^2 - 2x + 16\]Multiplying by -2,\[\large\rm 0 = x^2+4x-32\]Factors into,\[\large\rm 0=(x-4)(x+8)\]So if we want the arm to reach the object in 5 seconds instead of 4,
we could replace this factor (x-4) with (x-5),\[\large\rm 0=(x-5)(x+8)\]And we would have to think about what that does to the original equation.

Answer 17

That seems a little convoluted though,
maybe that's not the approach they're looking for...
hmm

Answer 18

If in the quadratic we have \(\frac{2 - 7}{-1}\), then it would be 5 seconds

Answer 19

We can change the c term to 18 I think.

Answer 20

Ya you could do that maybe :)
So you're assuming that the b value is still -2,
and that the a is still -1/2.

So we need the square root to give us 49.

Answer 21

\[\large\rm 2^2-4\left(-\frac{1}{2}\right)c=49\]

Answer 22

I think we get 22.5 or something.. hmm

Answer 23

@zepdrix i agree you - 5 need being a root of this quadratic so this mean that (x-5)(x-8)
will result sure this quadratic

Answer 24

There are many changes you could make to get a root of 5 though,
so this route should be fine as well :)

Did you follow what I was saying?
In order to get that 7 we must have,\[\large\rm b^2-4ac=49\]

Answer 25

yes

Answer 26

Ya I like your approach better xy,
it's a much simpler way of altering the problem.

You simply replace increase the 16 by 6.5 more to get the root x=5 that we're looking for.

Answer 27

replace the 16 by 22.5*
or increase the 16 by 6.5*

blah my words got mixed up there.

Answer 28

Okay so the quadratic is
\(f(x) = \frac{-1}{2}x^2 - 2x + 22.5\)

Answer 29

Yes, good.
Here it is graphed if you would like to verify the result.
https://www.desmos.com/calculator/bahqvckilk

Answer 30

Okay thanks! Can you help me on one more question please?

Answer 31

Sure!

Answer 32

An art teacher plans to spend a total of $100 on art supplies. The supplies can be purchased at a local art supply store or an online store. The actual teams are the same price in both the online store and local art shop. The Additional fees of sales tax, shipping, delivery, and processing vary between both stores.

Write two separate equations for each one of the stores with a total of $100.

Use the equations to explain why the art teacher should or should not order through the local art supply shop.

Answer 33

My equation for the local art supply shop:\(1.07x + $5 = $100\)
My equation for the online store: \(x + 0.2x = $100\)

Answer 34

Ooo looks great so far!

Answer 35

So do you think that I should find the value of x for the second part to the whole question?

Answer 36

The value of x for both of the equations.