the minimum vertical distance between the parabolas 1 and y x-x? y = x2 7. Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible. ( 8. Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible. 13. 9. A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is kN 14. Y = 1 + N2 Where k is a positive constant. What nitrogen level gives the best yield? 15. . The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function 16. 100I P = 12 1 1+ 4 where I is the light intensity (measured in thousands of foot- candles). For what light intensity is P a maximum? 17 11. Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four 18 pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Intro- duce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that 1 relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a). 12. Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bend- VIer

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.