In the most general sense, a value is any Real-number. Here are some examples of Real values:
\(\displaystyle 1,024 \quad -55 \quad \dfrac{69}{7} \quad 11^{2} \quad 9.8\overline{571428} \quad \sqrt{43} \quad \pi \quad 6.022 \times 10^{-23} \quad 0.3333…\)
… When you say finding y and finding x, I'm visualising the coordinates like the centre (h,k) which is (x,y) …
Yes, your parabola is composed of an infinite number of points. (In fact, between
any two points on the parabola that are as close together as you can imagine, there are still an infinite number of points between them! Individual points are infinitely microscopic, so they don't exist as real objects. They exist only in our mind; they're a very useful concept.)
Each point has an address which takes the form of an ordered pair of numbers (x,y) called coordinates. The symbols (h,k) represent the coordinates of the vertex point. The vertex of a parabola is the turning point of the curve. (You used 'centre', and it's true that the vertex is centered horizontally, but the vertex is not centered vertically. So, in the US, we prefer the name 'vertex'.)
Part of your goal is to determine the coordinates of the point on the parabola which is vertically above each end of the pool width. The y-coordinate of these points gives the height of the roof above the water -- at each end of the pool width.
… What methods am I suppose to use? How do I do them? …
These are good questions for you to ask your instructor. I don't even know what class you're taking, so I'm unable to answer these questions.
Also, you posted your exercise on the Calculus board, but it's not a calculus exercise. It's algebra. You mentioned that you've been doing implicit differentiation. That seems peculiar because (in my view) it is backwards to study calculus methods without first learning basic algebra.
Here's how I would do the exercise. Introduce a coordinate system on the given diagram, and label the known points with their coordinates.
Do the labeled points make sense? For examples: at the point where the roof meets the pool deck (to the right), the coordinates are (18,0). The x-coordinate measures horizontal distance from the vertical axis; so (18,0) is 18 meters away from the center of the pool's width (to the right). The point (-18,0) also measures horizontal distance, but in the opposite direction (to the left). The y-coordinate measures vertical distance from the horizontal axis; so (18,0) is 0 meters above the pool deck. The vertex point has coordinates (0,8.5). The x-coordinate is 0, so the point is zero meters away from the vertical axis. The y-coordinate is 8.5, so the point is 8.5 meters above the horizontal axis (i.e., the water's surface).
The horizontal distance from (-18,0) to (18,0) is 36 units. This matches the requirement that the width of the roof (at the pool deck) is no more than 36 meters.
The pool width is given as 18 meters. The vertical axis runs through the center of the width (cutting it in half), so the points at each end of the pool width must each be 9 meters away from the vertical axis. That's why the end points have coordinates (-9,0) and (9,0).
All parabolas that open either upward or downward (like yours) are graphs of what we call 'Quadratic Equations'. These equations may be expressed in various forms; I used this generic form:
y = a∙x^2 + b∙x + c
where symbols a,b,c each represent some fixed, Real number. For some arbitrary parabola, the values of a,b,c could each turn out to be
any Real number, with
one exception: symbol a cannot equal zero. If symbol a
were equal to zero, then the term a∙x^2 would go away, and the resulting equation would be y=b∙x + c. But that's not quadratic; it's linear (its graph is a straight line, not a parabola).
If we know the coordinates of three points on a parabola, then we can find the values of a,b,c using an algebraic method known as "writing and solving a system of equations". It goes like this.
One at a time,
substitute the x- and y-coordinates from a known point into the generic form and simplify (that is, replace all symbols x in the generic form with its Real value, and replace symbol y with its Real value.) Doing this gives a new quadratic equation containing only the symbols a,b,c (no symbol x and no symbol y).
Point #1: (-18,0)
0 = a∙(-18)^2 + b∙(-18) + c
Simplify:
0 = 324∙a - 18∙b + c
Point #2: (18,0)
0 = a∙(18)^2 + b∙(18) + c
Simplify:
0 = 324∙a + 18∙b + c
Point #3: (0,8.5)
8.5 = a∙(0)^2 + b∙(0) + c
Simplify:
8.5 = c
The three equations in red comprise our "system of equations". Did you notice? We just found the value of c.
Therefore, the Quadratic Equation we're looking for can now be written as:
y = a∙x^2 + b∙x + 8.5
We continue solving the rest of the system, to find the values of symbols a and b.
We can add equations, to get a new, valid equation. Let's add the first two (red) equations, in our system:
0 = 324∙a - 18∙b + 8.5
0 = 324∙a + 18∙b + 8.5
---------------------------
0 = 648∙a + 0∙b + 17
Simplify:
0 = 648∙a + 17
See what happened? -18∙b plus 18∙b combined to yield 0∙b because -18+18 is zero (opposites cancel each other out, when added). This is why I chose to add the first and second equations; I could see that symbol b would be eliminated from the result. Now we have an equation that contains only one symbol (a), so we can solve the equation and find the value of a.
We solve the equation 0 = 648∙a + 17 using two algebraic steps: (1) subtract 17 from each side, and (2) divide each side by 648.
(1) 0 - 17 = 648∙a + 17 - 17
Simplify:
-17 = 648∙a
(2) -17/648 = 648∙a/648
Simplify:
-17/648 = a
Now we know the value of a. We're getting closer; we need to find b:
y = -17/648∙x^2 + b∙x + 8.5
Let's go back to the second equation in our system:
0 = 324∙a + 18∙b + c
We have the values for symbols a and c, so we substitute them into this equation and simplify. That will yield a new equation containing only symbol b, so we can solve for b:
0 = 324∙(-17/648) + 18∙b + (8.5)
Simplify:
0 = -17/2 + 18∙b + 8.5
0 = -8.5 + 18∙b + 8.5
Opposites cancel when added, yes? Therefore, -8.5 + 8.5 is 0:
0 = 18∙b
We see that b must be zero.
We have found the Quadratic Equation whose graph is your parabola:
y = -17/648∙x^2 + 8.5
Let's write the fraction -17/648 in decimal form, and round it to four places:
y = -0.0262∙x^2 + 8.5
Okay. This equation is a formula for finding y, if we know x. Symbol y also represents the height of the parabola above the horizontal axis, at any given value of x on the horizontal axis.
In other words, we can use the formula to determine the height of the roof above each end of the pool width.
At the right end: x = 9
Substitute x = 9, in the formula:
y = -0.0262∙(9)^2 + 8.5
Simplify:
y = 6.3778
The coordinates of the point on the parabola vertically above the right end of the pool width are (9,6.3)
The y-coordinate represents the height of the roof, remember? Therefore, the roof is about 6.3 meters above the water, at the right end of the pool width.
Due to a concept known as symmetry (the right half of the parabola is a mirror image of the left half), we know that the y-coordinate at the left end of the pool width is the same.
We also know that between the ends of the pool width the roof is even higher than at the ends. Hence, the entire roof is always at least 6 meters above the surface of the water.
This is just one method, to approach the exercise.
By the way, these boards do not comprise an online classroom. Volunteer tutors at this forum do not generally have the time or motivation to type up lessons or step-by-step solutions because free algebra lessons already exist at many other sites on the Internet. For examples, you could take a free, beginning algebra course at coursera.org or at khanacademy.com. You could also check out lesson sites, like purplemath.com or mathisfun.com We are mainly here to help students who already have some idea of what they're supposed to be doing, but get stuck at some step. We guide them past that point, and then they continue on their own to finish (or to get stuck again, in which case they show their work and ask another
specific question at that step). Please don't expect us to teach you algebra; I think you ought to enroll in a bonafide algebra course, before studying calculus (or whatever class you're currently taking).
Please read the forum's
submission guidelines. Cheers :cool:
PS: Oops, I did not properly round the value 6.3778 to one decimal place, above. I wrote 6.3, but it should be 6.4
I don't want to edit my diagram, so just replace 6.3 with 6.4, everywhere it appears.
Additionally, I ought to have labeled the point where the x-axis and y-axis meet as (0,0). We call that point "the Origin".
PPS: In this exercise, we must assume that the pool is filled to the brim (i.e., the water's surface is level with the pool deck). This is not realistic, unless that's how they do it in New Zealand. In the real world, if we needed to know the true height of the roof above the water (at any location), then we would need to know how far below the pool deck the water surface lies.
