david is taller than chris, chris is shorter than anne, anne is as tall as sarah, sarah is shorter than david. who is the shorte
2 answers:
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Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6g of the chemical and in the second experiment she uses 7.8 g of the chemical
(i)Given that the amounts of the chemical used form an arithmetic progression find the total amount of chemical used in the first 30 experiments
(ii)Instead it is given that the amounts of the chemical used for a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that the greatest number of experiments possible satisfies the inequality: and use logarithms to calculate the value of N.
Answer:
(a)963 grams
(b)N=17
Step-by-step explanation:
(a)
In the first experiment, Sarah uses 6g of the chemical
In the second experiment, Sarah uses 7.8g of the chemical
If this forms an arithmetic progression:
First term, a =6g
Common difference. d= 7.8 -6 =1.8 g
Therefore:
Total Amount of chemical used in the first 30 experiments
Sarah uses 963 grams in the first 30 experiments.
(b) If the increase is geometric
First Term, a=6g
Common ratio, r =7.8/6 =1.3
Sarah has a total of 1800 g
Therefore:
Sum of a geometric sequence
Therefore, the greatest possible number of experiments satisfies the inequality
Next, we solve for N
Changing to logarithm form, we obtain:
Therefore, the number of possible experiments, N=17
Answer:
The set is linearly dependent for every value of h.
Step-by-step explanation:
Recall that a set of vector {a,b,c,d} is a linearly dependent set of vectors if any of the vectors can be written as a linear combination of the other ones.
We are given the following vectors.a=(-1,4,6), b=(5,2,-3), c=(3,-5,-4), d=(12,-20, h). At first, we will check if a,b,c are linearly independent. To do so, we will calculate the following determinant (the procedure of the calculation is omitted).
Since the determinant is not zero, this implies that the vectors a,b,c are all linearly independent. Since a,b,c are all vectors in which is a 3-dimensional space, and they are 3 linear independent vectors, then they are automatically a base of this space. Consider now the vector d. Since {a,b,c} is a base of
, then it generates any vector of this space(i.e any other vector of the space is a linear combination of {a,b,c}). In particular, d. So the set {a,b,c,d} is linearly independent for any value of h