Consider the following problem: A box with an open top isto be constructed from a square piece of cardboard, 3 ftwide, by cutting out a square from each of the four cornersand bending up the sides. Find the largest volume that sucha box can have.(a) Draw several diagrams to illustrate the situation, someshort boxes with large bases and some tall boxes withsmall bases. Find the volumes of several such boxes.Does it appear that there is a maximum volume? If so,estimate it.(b) Draw a diagram illustrating the general situation. Introducenotation and label the diagram with your symbols.(c) Write an expression for the volume.(d) Use the given information to write an equation thatrelates the variables.(e) Use part (d) to write the volume as a function of onevariable.(f ) Finish solving the problem and compare the answerwith your estimate in part (a).
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.
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 bending up the sides. Find the largest volume that such
a box can have.
(a) Draw several diagrams to illustrate the situation, some
short boxes with large bases and some tall boxes with
small bases. Find the volumes of several such boxes.
Does it appear that there is a maximum volume? If so,
estimate it.
(b) Draw a diagram illustrating the general situation. Introduce
notation and label the diagram with your symbols.
(c) Write an expression for the volume.
(d) Use the given information to write an equation that
relates the variables.
(e) Use part (d) to write the volume as a function of one
variable.
(f ) Finish solving the problem and compare the answer
with your estimate in part (a).
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