Learning Resources
Outcomes
By the end of this lesson you will be able to answer questions like the ones in the pre-test below.
Approach #1
Answer the questions on the pre-test below in your Math 3103 binder. Then refer to the pages of the lesson that contain instruction on the questions you were unable to do, for which you got an incorrect answer, or for which you would like some further explanation.
Approach #2
Proceed directly to the first page of the lesson without attempting the pre-test. To do this simply click on the Lesson button above. You can then return to this pre-test later and use it as a worksheet.
Pre-test
1. Water is being poured into a right circular conical tank as shown in the diagram below:
(a) Express the volume of the water in the tank at any given time as a function
of the depth of the water, d , and the radius of the tank at the surface of
the water , r .
(b) If the tank has a total depth of 250 m and a radius of 50 m, express the
volume of water in the tank as a function of d only. (Hint: draw in lines for
the total depth and radius, and use similar triangles to find a formula relating
h and d).
(c) If water flows into the tank at a rate of 5000 m3 per minute, what is the depth of the water after (i) 15 min, (ii) 30 min , (iii) 1 hour
2. A plane leaves Deer Lake and travels north at the rate of x km/h. At the same time, a second plane leaves Deer Lake and travels east at the rate of y km/h. (Hint: sketch a diagram to help answer the following questions)
(a) Express the distance between the planes, d , as a function of x, y ,and the time, t, since they have left.
(b) If the second plane is traveling 50 km/hr faster than the first, express d as a function of x and t.
(c) If the first plane is traveling 300 km/h, how far apart are they after 2 hours?
3. A rectangular piece of sheet metal is x metres wide by y metres long:
It has congruent squares with sides z metres cut from the corners:
The edges are then turned up along the red lines to make an open box with no top.
(a) Express the volume of the box, V , as a function of x , y , and z.
(b) If the length is 7 m more than the width, express V as a function of x and z.
(c) If the width of the piece of metal is 8 m, what are the dimensions of the square that has to be cut out to give the box a volume of 88 m3 ?
Students will be expected to:
- Rearrange formulas to solve for any variable.
- Simplify problem situations involving functions of two or more variables to functions of one variable.
Prerequisites
To be successful in this lesson, it would be helpful to know the following:
- How to rearrange formulas and solve for any given variable.
- The formulas for the surface area and volume of various geometric solids.
- Pythagorean Theorem.
- Similar triangles
- Formula relating distance, rate, and time.
- How to substitute an expression into an equation.