Find two positive numbers satisfying the given requirements. The product is 432 and the sum of the first plus three times the second is a minimum.

Step-by-step explanation:

Let the two positive integers be x and y.

If their product is 432, then

xy = 432 1

Also if the sum of the first plus three times the second is a minimum, then;

p(x) = x + 3y

From 1;

y = 432/x 3

Substitute 3 into 2;

p(x) = x+3y

p(x)= x + 3(432/x)

p(x) = x + 1296/x

Since the expression is at minimum when dp(x)/dx = 0

dp/dx = 1 + (-1296)/x²

dp/dx = 1  -1296/x²

0 =  1  -1296/x²

0 = (x²-1296)/x²

cross multiply

0 = x²-1296

x² = 1296

x = √1296

x = 36

Since xy = 432

36y = 432

y = 432/36

y = 12

Hence the two positive numbers are 36 and 12

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.