Question

Please help! Will FAN AND MEDAL!

For what value of k, k > 0, does the integral from k to 0 of the quantity 3 times k minus 2 times k times x, dx equals k squared.

a. 1
b. 2
c. 3
d. 4

Answer 1

Can you please write out the equation, I can assist you from there.

Answer 2

\[\int\limits_{k}^{0}(3k-2kx)dx=k^2\]

Answer 3

@xGuardians

Answer 4

\[\int_k^0 (3k-2kx) \ dx = k^2 \\
= \int_k^0 3k \ dx - \int_k^0 2kx \ dx \\
= 3k\int_k^0 dx - 2k\int_k^0 x \ dx = k^2
\]
Can you solve this equation for \(k\)?

Answer 5

which part of the equation should I put into my calculator?

Answer 6

First, you need to evaluate the integrals.

Answer 7

evaluate what integral?

Answer 8

Let's back up, do you understand how I got
\[
3k\int_k^0 dx - 2k\int_k^0 x \ dx = k^2
\]

Answer 9

Yes I understand your steps above

Answer 10

Okay, so now we can go one step further and evaluate the integrals like so:
\[
= 3k \bigg[ x \bigg]_k^0 - 2k \bigg[ \frac{x^2}{2} \bigg]_k^0 = k^2
\]

Does this make sense?

Answer 11

Yes it does

Answer 12

@andrewyates

Answer 13

So now,
\[
= 3k(0-k)-2k(\frac{0^2}{2} - \frac{k^2}{2}) \\
= -3k^2+k^3
\]

Answer 14

Which is equal to \(k^2\):
\[
-3k^2+k^3 = k^2 \implies \\
k^3-4k^2 = 0 \implies \\
k^2(k-4) = 0 \implies \\
k = 0, 4
\]

Answer 15

Okay but that is not one of my answers?

Answer 16

@andrewyates

Answer 17

k = 4 is one of your answers. D.

Answer 18

ohh haha I was going with 0,4