I have been working on this problem for 30 minutes and have gotten no where! I need to find the maximun area of a norman window ( a rectangle with a semicircle on top) when the perimeter is 30 ft. all i kno wis that the base os the semicircle is equal to the length of the rectangle. HELP :shock:
maximum area
- Thread starter frichar3
- Start date
- Joined
- Feb 4, 2004
- Messages
- 16,583
Draw the Norman window, labelling the width as "w", the height of the rectangular part as "h", and the radius of the half-circle part as "r". Notice that r = w/2.
To find the "perimeter" of the half-circle part, plug "w/2" into the circumference formula for a circle, and multiply the result by 1/2 (since you're only taking half the circle).
To find the "perimeter" of the rectangle part, add the one width and the two heights. (The other width is where the rectangle joins the half-circle, so it doesn't count as "perimeter".)
Add these two expressions, in terms of w and h, and set equal to "30". Solve for one of the variables. (It doesn't really matter which, though you may find one or the other easier to work with.)
By solving, you have an expression for one of the variables (suppose "w") in terms of the other variable (which then would be "h").
Now use the area formula for a circle (and multiply again by 1/2) and the area formula for a rectangle. Substitute for the one variable by using that expression you derived. Adding these two formulas together, you have a formula for the total area, in terms of only one variable.
Maximize.
If you get stuck, please reply showing how far you have gotten. Thank you.
Eliz.
To find the "perimeter" of the half-circle part, plug "w/2" into the circumference formula for a circle, and multiply the result by 1/2 (since you're only taking half the circle).
To find the "perimeter" of the rectangle part, add the one width and the two heights. (The other width is where the rectangle joins the half-circle, so it doesn't count as "perimeter".)
Add these two expressions, in terms of w and h, and set equal to "30". Solve for one of the variables. (It doesn't really matter which, though you may find one or the other easier to work with.)
By solving, you have an expression for one of the variables (suppose "w") in terms of the other variable (which then would be "h").
Now use the area formula for a circle (and multiply again by 1/2) and the area formula for a rectangle. Substitute for the one variable by using that expression you derived. Adding these two formulas together, you have a formula for the total area, in terms of only one variable.
Maximize.
If you get stuck, please reply showing how far you have gotten. Thank you.
Eliz.