The profit P(t) of a startup company, in thousands of dollars,...

Question

The profit P(t) of a startup company, in thousands of dollars, t months after receiving its initial funding, can be modelled by the equation:
P(t) = –0.2(t^2)+ k(t) – 15, where k is a positive constant.

During the first 24 months after funding, was the company's profit equal to $m thousand at most once?

1. m < –15

2. k < m

KaranATS · Answer

hey, i cant seem to make any headway on how to carry on here, especially within a reasonable time limit. can someone help?

repellatquas · Answer

We have been fed with the value of t = 24 and asked whether P(24) = m at most once (explained below).

P(24) = -0.2(24^2) + k(24) - 15 = -130.2 + 24k

Now, we need to find out whether -130.2 + 24k = m for only one value of k.

Option B does not tell us anything about m. Eliminate.

Option A. m < -15
Try values of k =1,2,3,4,5,6

1. For k>=6, LHS becomes positive and can never be equal to m (a negative value, m<-15) {Irrelevant}
2. For k<=0, k can't be negative as necessited by the question
3. For k=1,2,3,4,5 : LHS can be equal to m (assuming the values found in LHS=m)

Thus, by A we know for sure that various values of k result in the profit being equal to m. Hence, profit can be equal to m multiple times.

SomethingSomw · Answer

I couldnt exactly understand the approach repellatquas went with, but i had a separate idea to deal with this question.

We know that the equation P(t) = –0.2(t^2)+ k(t) – 15 ;___(1) is a parabola. and because the coefficient of t^2 is negative, we know that the parabola is downwards with X axis as time, and Y axis as profit (P).
For us to know whether the value of $m thousand is only hit once or never during 0 to 24 months, we need to find the intersection of the line P(t) = m ;___(2) and the equation for profit. If there are multiple solutions to that equation then we need to see whether the solutions lie in the range of 0 <= t <= 24. Fortunately didnt have to go as far for the given case.

To find solutions of the equations (1) and (2), we need to solve for m = –0.2(t^2)+ k(t) – 15
=> –0.2(t^2)+ k(t) – (15+m)
for a quadratic equation of the form ax^2 + bx + c = 0, the number of solutions can be found by seeing whether b^2 - 4ac is positive (2 solutions), negative (0 solutions) or zero (1 solution)

So, now we are trying to find whether k^2 - (-0.2)(-15-m) > 0?
=> k^2 > 0.2(15+m) ? ___(3)

According to option 1, m < -15; which means m+15<0 which will make the above equation false, so we have our answer that the value of profit m cant be hit multiple times since there are no solutions for their intersection => A option is so far so good

According to option 2, k < m; and k has to be positive according to the original statement. So now m cant be negative, and we cant say for a fact whether the equation (3) is true or not as both LHS and RHS are positive and could have different values for k and m such that the inequality is satisfied/not satisfied.
Therefore, option 2 cant give us a concrete answer.

=> A OPTION is correct

aditya khanna · Answer

maximum of the quadratic equation –0.2(t^2)+ k(t) – 15=0 , lies on -b/2a which is k/0.4 
after substituting t = k/0.4 , we get 
Pmax = k^2 / 0.8 -15 
if the value of m is less than Pmax , then Statement 1 is sufficient as there will be multiple m values 
given  m < -15 , then certainly m will be less than Pmax , hence Statement 1 is sufficient.

Edskore · Answer

Key concept being tested: Data Sufficiency with Quadratic Functions — specifically, understanding when a downward-opening parabola can intersect a horizontal line at most once within a given domain.

Setup: P(t) = -0.2t^2 + kt - 15, where k > 0. The parabola opens downward (coefficient of t^2 is negative). We want to know: during t in [0, 24], does profit equal $m thousand at most once?

Setting P(t) = m gives us: -0.2t^2 + kt - (15 + m) = 0, or equivalently 0.2t^2 - kt + (15 + m) = 0.

The key insight: use the product of roots from Vieta's formulas.

1. For this quadratic in t: product of roots = (15 + m) / 0.2 = 5(15 + m).

2. Evaluate Statement (1): m < -15.
If m < -15, then 15 + m < 0, so the product of roots = 5(15 + m) < 0.
A negative product of roots means one root is negative and one is positive.
The negative root is outside our domain t >= 0, so within [0, 24], P(t) = m at exactly one t-value (or zero times if the positive root > 24).
Either way, "at most once" — Statement (1) ALONE is SUFFICIENT.

3. Evaluate Statement (2): k < m.
This tells us m > k > 0, so m is positive. The maximum value of the profit function is P_max = k^2/0.8 - 15. Statement (2) does not pin down k or m precisely enough to determine whether the line y = m crosses the parabola once or twice within [0, 24]. INSUFFICIENT.

Answer: A

Common trap: Many students try to analyze the vertex location or plug in numbers for k, rather than using Vieta's product-of-roots shortcut. The moment you write the equation P(t) = m in standard form, checking the sign of the product of roots immediately tells you whether both solutions are on the same side of zero.

Takeaway: On DS questions with quadratic equations, Vieta's formulas (sum and product of roots) can often settle sufficiency faster than solving directly.

The profit P(t) of a startup company, in thousands of dollars,...

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GMAT Data Sufficiency (DS) Questions

The profit P(t) of a startup company, in thousands of dollars,...

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