0.10 Simultaneous equations

to yield

This equation can be simplified as

Combining terms yields

The solution for the variable B is 100. So we find that the mass of the second alloy is 100 grams. Because the sum of the masses of the two alloys is 300 grams, the mass of the first allow is 200 grams.

Photovoltaic arrays

A photovoltaic system is a system which uses one or more solar panels to convert sunlight into electricity. It consists of multiple components, including the photovoltaic modules, mechanical and electrical connections and mountings and means of regulating and/or modifying the electrical output.

Due to the low voltage of an individual solar cell (typically on the order of 0.5V), several cells are wired in series in the manufacture of a "laminate". The laminate is assembled into a protective weatherproof enclosure, thus making a photovoltaic module or solar panel. Modules may then be strung together into a photovoltaic array. The electricity generated can be either stored, used directly (island/standalone plant)or fed into a large electricity grid powered by central generation plants (grid-connected/grid-tied plant) or combined with one or many domestic electricity generators to feed into a small grid (hybrid plant)

A photovoltaic array (or solar array ) is a linked collection of solar panels. The power that one module can produce is seldom enough to meet requirements of a home or a business, so the modules are linked together to form an array . Most arrays use a device known as an inverter to convert the DC power produced by the modules into alternating current that can be used to power lights, household appliances, motors, and other loads. The modules in a photovoltaic array are usually first connected in series to obtain the desired voltage. Once this is accomplished, the individual strings are then connected in parallel to allow the system to produce more current.

Example 6: A photvoltaic array is to be installed. It is a requirement that the length of the array be 8 inches longer than its width. A second requirement states that the total area of the array be 180 in ² .

Find the length ( L ) and width ( W ) of the array.

We may use the first requirement to construct the first simultaneous equation

The second requirement contributes a second simultaneous equation

We may substitute the expression for L given by equation (18) into equation (19)

or equivalently

This quadratic equation can be factored as

The two roots for W are 10 in and -18 in . Because it is physically impossible to have a negative value for width, we ignore the second root and conclude that the width of the photovoltaic array is 10 in. We use this value to solve for the length (L) of the array

Therefore, the dimensions of the photovoltaic array are 18 x 10 in.

Exercises

1. A rowing team competes in a tournament that has two legs. The first leg covers 15 miles in the direction of the current. The second leg covers the same 15 miles, but against the current. It takes 25 minutes to complete the first leg and 30 minutes to complete the second leg. Assuming that there is no current present, what would be the speed of the boat? What is the speed of the current?
2. A beam of length 3 meters is supported by a fulcrum. Force is applied at each end of the beam. Assume that the force ( F ₁ ) on the left hand side of the beam is 6 N while the force on the right hand side of the beam ( F ₂ ) is 7 N. If the beam is static, then what are the lengths between the each end of the beam and its fulcrum?
3. Suppose that we are interested in implementing a computer algorithm on three processors. Let us call these processors A and. The steps involved in the accomplishment of the algorithm are divided among the three processors. In all there are 36,000,000 computations which will need to be performed on the three processors. In performing the overall task, processor A is in service for 2 seconds while the processor B is in services for 4 seconds. A second algorithm is implemented in parallel on processors A and B. In it, a total of 23,000,000 computations are required. In performing the task, processor A is in service for 1 second and processor B is in service for 1.5 seconds. Find the processing rate (computations/s) for each processor.
4. One alloy is made up of a concentration of 25% copper and 75% manganese. A second alloy consists of a concentration of 50% copper and 50% manganese. The two alloys are melted and re-combed. The resulting mixture weighs 450 gram and consists of 35% copper and 65% zinc. Find the weights of the original alloys.