You want to build a storage shelf in the garage. It is to be used for paints, oil for the car, bleach, laundry detergent, etc. You decide that you want it to be a foot wide and ten feet long. This way you can store two rows of larger items such as bottles of bleach or cans of paint. First we need an estimate for the total weight it is expected to hold. A gallon of water weighs eight pounds. A bleach bottle or a can of paint is about 9 inches wide. So, we might have a total weight of (10 ft x 12 in / 9in) x 2rows x 8 lb = 210 lb approximately. The shelf will be rest on a runner screwed securely into the wall. The outside edge of the shelf is to be supported with a nylon cord that is safety-rated to a tension of 15 lb. The twine will hang vertically downward from screweyes in the ceiling. How many strings should you use?
You've decided to build a new deck. It is to be 10 ft wide and run the 30 ft length of the house. You must estimate the weight of the deck, as well as, the maximum load you expect it to support. That means you need an estimate of the weight of the lumber and how many people you expect at your deck party. A reasonable value might be around 6,000 lb. For now, lets denote the total load as W, assumed to be spread uniformly over the deck. The deck is to be supported where it is attached to the house and at the outside edge by support beams that make a 45o angle as shown. What will be compressive tension in the beams?
Installing new cabinents can really spruce up the look and functionality of your kitchen. But what kind of attachments are necessary? A standard method of installation is to attach a running support strip along the wall for the bottom edge of the cabinents to rest upon. The bottom strip supports the weight of the cabinets. The cabinents are held against the wall with screws at the top of the cabinents that screw directly into the wall. (For typical wood frame and drywall construction, it is very important to make sure the screws go into studs and not just the drywall. For concrete walls, you will need to install something like anchor bolts.) Assume the cabinets are 1 ft deep by 3 ft high and will weigh 200 lb (uniformly distributed) when full of dishes. What total horizontal force is required of the top screws? What force is supplied by the bottom strip? For the bottom strip, break the force into two forces, one horizontal and the other vertical as shown.
Look at the diagram. Using the hints given, we draw all the forces acting. We take the pivot point at the bottom of the rafter. Since we have an equilateral triangle, the moment arms are easy to figure out. Directly below the point where the two rafters meet must be exactly half the distance, or 4 ft along the joist. So the line of action of the weight (acting at the center of the rafter) must be 2 ft. Using the Pythagorean theorem, the line of action of the force at the top must be 
(82 - 42) = 6.9 ft. Noting the directions, the moments equation gives us 
How do you find the perpendicular distance? It depends upon your level of mathematical expertise. (You can always lay sticks and strings out on a table and measure it directly.) The length of the cord from the Pythogorean Theorem is the square root of 12 + 32 = 3.16. If you know your trigonometry, you can use sines and cosines. But here an interesting geometry trick. The area of a triangle is one-half of the base times the height. Taking the 1 ft shelf as the base and the 3 ft as the height, we get 1.5 ft2. Now take the hypotenuse (the cord) as the base and d as the height. So 1/2 (3.16)d = 1.5 ft2, and d = .95 ft or about 11 inches. Plugging into the moments equation, the new tension will be 110 lb. So we'll need 7.4 cords, instead of 7. Not a big difference if we plan to double for the safety factor anyway. However, as the wall attachment of the cord is moved closer to the shelf, the tension needed grows quickly. If we attach the strings at only 1 ft above the shelf, the tension needed is 40% greater than before. At 6 inches, the tension is over 220 lb (> 200% increase!)