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question 10 of 41 mscs district benchmark 2 - algebra 1 which statement…

Question

question 10 of 41
mscs district benchmark 2 - algebra 1
which statements are true? select three that apply.
□ after 10 years, one-half of the substance will remain compared to its initial value.
□ the maximum amount of the substance is 3,500 grams.
□ the value of a decreases as the value of t increases.
□ over equal time intervals, the value of a changes by the same number of grams.
□ the amount of substance lost between 1 and 2 years is equal to approximately half the amount of substance lost between 0 and 1 year.

Explanation:

Brief Explanations

Assuming this is about exponential decay (common in Algebra for half - life problems):

  1. "After 10 years, one - half of the substance will remain compared to its initial value" – In half - life, after each half - life period, half the substance remains. If the half - life is 10 years, this is true.
  2. "Over equal time intervals, the value of a changes by the same number of grams" – No, in exponential decay, it changes by a percentage (or multiplicative factor), not a constant number of grams.
  3. "The amount of substance lost between 1 and 2 years is equal to approximately half the amount of substance lost between 0 and 1 year" – In exponential decay, the amount lost decreases over time (since it's a percentage of the remaining amount). So the amount lost in the first year is more than half of the amount lost in the second year. Wait, maybe I got this reversed. Wait, if at year 0, amount is \(A_0\). At year 1, \(A_1 = A_0\times r\) (r is the decay factor, \(0\lt r\lt1\)). Amount lost from 0 - 1: \(A_0 - A_1=A_0(1 - r)\). At year 2, \(A_2 = A_1\times r=A_0\times r^{2}\). Amount lost from 1 - 2: \(A_1 - A_2=A_0r(1 - r)\). Now, \(\frac{A_0r(1 - r)}{A_0(1 - r)}=r\). If \(r = 0.5\) (half - life), then the amount lost from 1 - 2 is half of the amount lost from 0 - 1. So this statement is true.
  4. "The maximum amount of the substance is 3,500 grams" – If it's decaying, the initial amount could be the maximum, so if initial is 3500, this is true.
  5. "The value of 4 decreases as the value of t increases" – Assuming the formula is \(A(t)=A_0r^{t}\), as t increases, A(t) decreases (since \(r\lt1\)), so if A(t) is the amount and t is time, and 4 is a mis - label (maybe A(t) when t = 4), then as t increases, the amount decreases. But the wording is odd.

Wait, maybe the correct statements are: "After 10 years, one - half of the substance will remain compared to its initial value", "The maximum amount of the substance is 3,500 grams", and "The amount of substance lost between 1 and 2 years is equal to approximately half the amount of substance lost between 0 and 1 year" (depending on the decay model). But since the question says to select three, and given the options, likely the three true ones are: "After 10 years, one - half of the substance will remain compared to its initial value", "The maximum amount of the substance is 3,500 grams", and "The amount of substance lost between 1 and 2 years is equal to approximately half the amount of substance lost between 0 and 1 year" (or maybe the first, fourth, and fifth? Wait, the fifth says "The value of 4 decreases as the value of t increases" – maybe a typo, like "The value of A (amount) decreases as t (time) increases", which is true for decay.

But since the problem is in Algebra (subfield of Mathematics), and it's about exponential functions (decay), let's re - evaluate:

  1. "After 10 years, one - half of the substance will remain compared to its initial value" – True (half - life concept, exponential decay).
  2. "The maximum amount of the substance is 3,500 grams" – True (initial amount is maximum in decay).
  3. "The amount of substance lost between 1 and 2 years is equal to approximately half the amount of substance lost between 0 and 1 year" – True (as shown in the decay formula, with half - life, the amount lost in each subsequent year is half of the previous year's loss).

The other two: "Over equal time intervals, the value of a changes by the same number of grams" – False (exponential change is multiplicative, not additive). "The value of 4 decreases as the value of t increases" – Wording is u…

Answer:

  1. After 10 years, one - half of the substance will remain compared to its initial value.
  2. The maximum amount of the substance is 3,500 grams.
  3. The amount of substance lost between 1 and 2 years is equal to approximately half the amount of substance lost between 0 and 1 year.