Given the points (7,3) ( 7 , 3 ) and (k,−7) ( k , − 7 ) , for which values of k would the distance between the points be 2√ 41 ?

You have to use the distance formula here to figure this one out. The distance formula is

.

We have the distance, we have one of the x-coordinates and both of the y coordinates so we fill in accordingly:

which simplifies to

. What we did there was expand the binomial of (k-7)(k-7). That simplifies even further to

.

Now we have to find a way to solve that for k, which means to get it out from under the square root sign. Since squaring a square root "undoes" the square root AND we have square roots on both sides of the equation, we will square both sides of the equation (which you would have to do anyway since this is an equation and if you do one thing to one side of it you have to do the same thing to the other side!) Squaring both sides,

. Since squaring undoes the square roots we now have

. Multiplying on the left side and then adding that product to the right side and setting the whole mess equal to 0 gives us

.

Now we just have to factor that in whatever way we deem the most efficient and effective. We can figure what 2 numbers that add together to equal -14 also multiply together to equal -15. Those 2 numbers are k=15 and k=-1. In factorization form that is (k-15)(k+1). Do the FOILing just to see that it's correct and you're good to go!

it is true

step-by-step explanation:

all the combinations add up i used   calculator

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.