Please Help I do not understand this college algebra word problem?

Question

If someone could please help me answer this with the steps so I can learn how to write this out I will give you best answer, and a virtual hug, lol.

The sum of the base and the height of a parallelogram is 69 cm. Find the dimensions for which the area is a maximum.

Answer 1

The area of a parallelogram is equal to the base times the height.
A = (b)(h)

The way to get maximum area is a square (which is a parallelogram). In a square, all sides are the same. Since 69/2 + 69/2 = 69...

A = (69/2)(69/2) = 4761/4 = 1190.25 cm²

Answer 2

If x is the base of the parallelogram, then 69-x is the height. The area of a parallelogram is base*height, so the area of our figure is A=x(69-x).

If you distribute the x and arrange in descending order, you get A=-x^2+69x, which is a quadratic function. If you know calculus, you can take the derivative to find the maximum. If you don't know calculus, you can still find the max.

Since the formula for area is a quadratic with a negative coefficient of x^2, the graph would be a parabola opening down. Therefore the maximum would be at the vertex. You can use the formula x=-b/2a to find where the vertex is. This will give you a value for x, which is the length of the base. Then you will be able to find the area by plugging that back into the formula.

Good luck!

Answer 3

Since this is an ALGEBRA problem, the solution should be sought in algebraic terms.

Work with lengths in cms, areas in cm^2.
Let base = x, then height = 69 - x. (They sum to 69).
Then area A = x(69 - x) = - (x^2 - 69x).

Now complete the square in the (negative!) parentheses,
correcting for the newly introduced term:

Then A = - (x^2 - 69x +[34.5]^2) + (34.5)^2

= - (x - 34.5)^2 + (34.5)^2

Clearly, the maximum of this function occurs when x = 34.5
when the subtracted term is zero. (For any other value of x, you'd be subtracting a square and therefore positive term.)

So the area A is maximum for base = height = 34.5 cm, and the maximum area itself (though not requested) is (34.5)^2 cm^2, i.e. 1190.25 cm^2

Note that calculus is QUITE UNNECESSARY for problems involving quadratic functions like this. Spotting how to complete the sqare will work every time.

Live long and prosper.

Answer 4

the formula for area of a parallelogram is b*h so if the sum is 69 find the two numbers and multiply them. 60 and 9, then times them 60*9. and repeat until you find the biggest. so you should start in the middle 35 and 34, then go to 36 and 33, 37 and 32, go until you can't get any higher.

Answer 5

Let x = pounds of $3/pound coffee x + 60 = pounds of $4/pound coffee after mixing 4(x + 60) = 3(x) + 6(60) 4x + 240 = 3x + 360 4x - 3x = 360 - 240 x = 120 pounds of $3/pound coffee

Answer 6

maximum will be base=height=69/2

the steps are:
let base be x,
so height is 69-x
and area is x(69-x)=69x-x^2

Take derivative and set it equal to zero:
69-2x=0
x=69/2

Answer 7

i guess it is 1190 cm^2

Answer 8

may 79