can anyone help me with a maths question?

Question

hi i have a question that says that:

t= p/(4d-a) and i am told that p=6.9, d=3.4 and a=5.

a) find the lower bound of t
b) find the upper bound of t

what does lower/upward bound mean?

Answer 1

Given that p is 6.9 to one decimal place, the exact value will be in the range 6.85 to 6.95 (there is room for discussion about using 6.94999... as the upper bound but this is irrelevant here!).
For d = 3.4 (1dp) we have the range 3.35 to 3.45.
For a = 5 (nr whole) we have the range 4.5 to 5.5.

a) The lower bound of t is found by minimising the numerator and maximising the denominator on the RHS of the formula.
Thus, we take p = 6.85, d = 3.45 and a = 4.5.
So that t = 6.85/(4*3.45 - 4.5) = 0.7 (1dp)

b) The upper bound of t is found by maximising the numerator and minimising the denominator on the RHS of the formula.
Thus, we take p = 6.95, d = 3.35 and a = 5.5.
So that t = 6.95/(4*3.35 - 5.5) = 0.9 (1dp)

A very good example of how innocuous seeming roundings in measurements can effect calculations - here being a range of 28% of the lower bound.

Answer 2

A web search turned this up
--- begin quote ---
http://mathworld.wolfram.com/UpperBound.html

A function is said to have a upper bound if for all in its domain. The least upper bound is called the supremum. A set is said to be bounded from above if it has an upper bound.

Lower Bound

http://mathworld.wolfram.com/LowerBound.html

A function is said to have a lower bound if for all in its domain. The greatest lower bound is called the infimum.

--- end quote ----
In the case of your function, you have constant values for p, d, and a, so t would also be a constant and its upper and lower bound would also be this constant.
t = 6.9/((4x3.4)-5) = 6.9/8.6 ~= 0.8023
So for the values given, t has an upper bound and a lower bound of 6.9/8.6 or approx 0.8023
If p, d, and a were allowed to vary, this question would have a more complicated answer.
If p were allowed to increase without bound, then t would have no upper bound.
If p were allowed to be negative, then t would have no lower bound.
If 4d = a then the function would be undefined, and it would approach infinity as 4d approaches a. Negative or positive infinity depending on the sign of p and whether 4d is greater than a or less than a.

Oops, just read the next answer and it seems I may have misunderstood your question. If p = 6.9, etc, my answer would be correct. but if you mean that p can be in the interval from 6 to 9, the other answer would be correct.
At any rate, you would use the definitions I cited, to understand and find your upper and lower bounds.

Answer 3

The way to work out the lower bound is to put the values of p and d in the formula such that t takes it's smallest possible value. So putting p=6 and d=4 gives t=6/(4*4-5)=0.4

Similarly, to calculate the upper bound of t, we substitute the values of p and d which give the highest possible value of t. The variables in this case are p=9 and d=3, giving t=9/(4*3-5)=1.286

Note the way the variables were chosen so that, for the lowest bound, we had the lowest possible numerator (from a choice anywhere in the interval 6...9) and the highest possible denominator (from a choice in the interval 3...4). For the upper bound it was the opposite of this strategy (i.e make the numerator high while the denominator is made low).

Answer 4

you should use p 6 and d 3 and a 5 and take the number for t and than you should use p 9 d 4 and a 5 and take the number for t. these two numbers one is lower bound and other is upper bound. its easy but a bit tricky thats all