Question

What best can be said about the number of satellites serving C?

• A. Must be between 450 and 725
• B. Cannot be more than 800
• C. Must be between 400 and 800
• D. Must be at least 100
• A. 100
• B. 200
• C. 500
• D. 250
• A. At most 475
• B. Exactly 475
• C. At least 475
• D. No conclusion is possible based on the given information
• A. The number of satellites serving C cannot be uniquely determined.
• B. The number of satellites serving B is more than 1000 .
• C. All 1600 satellites serve B or C or S.
• D. The number of satellites serving B exclusively is exactly 250.

Answer 1

The Correct Option is C

The correct answer is (C): 

The number of satellites serving region C is given by:

\[ \text{Satellites serving C} = z + 0.3x + 100 + y \]

Substitute the values from the equations:

\[ z = 550 - 5y,\quad x = 5y + 250 \] \[ \Rightarrow \text{Satellites serving C} = (550 - 5y) + 0.3(5y + 250) + 100 + y \]

Simplify:

\[ = 550 - 5y + 1.5y + 75 + 100 + y = 725 - 2.5y \]

This expression is maximum when \( y \) is minimum. Minimum value of \( y = 0 \), so:

\[ \text{Maximum satellites serving C} = 725 - 2.5 \times 0 = 725 \]

From equation (3):

\[ z = 550 - 5y \] Since the number of satellites cannot be negative, we must have: \[ z \geq 0 \Rightarrow 550 - 5y \geq 0 \Rightarrow y \leq 110 \]

Now, the maximum possible value of \( y \) is 110.

When \( y = 110 \):

\[ \text{Satellites serving C} = 725 - 2.5 \times 110 = 725 - 275 = 450 \]

Therefore, the number of satellites serving C lies between 450 and 725.

Answer 2

Given the problem, we need to determine the minimum number of satellites serving B exclusively. Let's break down the information:

Let B = 2x, C = x, and S = x based on the ratio 2:1:1. 

The number of satellites serving all three (B ∩ C ∩ S) is 100.

Let B_e, C_e, and S_e be the satellites serving exclusively B, C, and S, respectively. Given C_e = S_e = 0.3B_e.

The number of satellites serving others (O) is the same as the number serving C and S but not B, let's denote this as C ∩ S - B.

Using these, we can set up equations based on the total number of satellites:

• Total satellites: B + C + S + O = 1600 (1)

• B = B_e + C ∩ B - S + C ∩ B ∩ S + B ∩ S - C (2)

• C = C_e + C ∩ B - S + C ∩ B ∩ S + C ∩ S - B (3)

• S = S_e + C ∩ S - B + C ∩ B ∩ S + B ∩ S - C (4)

• B ∩ C ∩ S = 100 (5)

• C_e = 0.3B_e (6)

• S_e = 0.3B_e (7)

Based on equation 6 and 7, suppose B_e = 1000, then C_e = S_e = 0.3 × 1000 = 300.

Hence, equations (2), (3), (4) become:

B = 2x = B_e + D

C = x = C_e + E

S = x = S_e + F

Substituting the known values: B = 2x = 1000 + D, C = x = 300 + E, S = x = 300 + F, where D = E = F = H, which satisfy total 1600 condition.

If D + E + F + O = 1600 - (B_e + C_e + S_e + B ∩ C ∩ S = 1900), so D + 100 + O = 600.

Thus, O = C ∩ S - B = C_e + S_e + 100.

If B_e is minimised to 250, solving directly from equilibrated conditions where B: 2x, hence deductions confirm B_e = 250 (since other conditions uphold a 4_ratio in disparity between C/S/B linked-exclusive constraints).

Satellite	Count
Bexclusive	250
Ce = Se	75
B ∩ C ∩ S	100

The minimum possible number of satellites serving B exclusively is confirmed as 250.

Answer 3

The correct answer is (A): 

We are given that at least 100 satellites serve 0 regions. So, we take:

\[ y \geq 100 \]

From earlier equations, the number of satellites serving S is:

\[ \text{Satellites serving S} = 0.3x + z + 100 + y \]

Substitute the known expression: \[ = 725 - 2.5y \]

This value is minimum when \( y \) is maximum. From constraint (3), maximum possible \( y = 110 \).

So, minimum number of satellites serving S: \[ = 725 - 2.5 \times 110 = 725 - 275 = 450 \]

This value is maximum when \( y = 100 \).

So, maximum number of satellites serving S: \[ = 725 - 2.5 \times 100 = 725 - 250 = 475 \]

Therefore, the number of satellites serving S is at most 475.

Answer 4

We are given relationships among the number of satellites serving regions B, C, and S. The goal is to find out the number of satellites serving at least two of B, C, or S. 

Definitions:

• $x$: Number of satellites serving only one region
• $y$: Number of satellites serving all three regions (B, C, and S)
• $z$: Number of satellites serving exactly two of B, C, or S
• Total satellites serving at least two regions = $2z + y + 100$ (as given)

Also given:

• $z = 550 - 5y$
• $x = 5y + 250$

Step 1: Total satellites serving at least two of B, C, or S

Total = $z + z + y + 100 = 2z + y + 100$

Substitute $z = 550 - 5y$:

$2(550 - 5y) + y + 100 = 1200$
$1100 - 10y + y + 100 = 1200$
$1200 - 9y = 1200$
$\Rightarrow y = 0$

Step 2: Find values of $x$ and $z$

If $y = 0$:

• $x = 5y + 250 = 250$
• $z = 550 - 5y = 550$

Step 3: Satellites serving region C

Let number of satellites serving C be $k$.
Given:

$k = z + 0.3x + 100 + y$

Substitute the values:

$k = 550 + 0.3 \times 250 + 100 + 0 = 550 + 75 + 100 = 725$

Step 4: Satellites serving region B

Given: Satellites serving B = $2k = 2 \times 725 = 1450$

Conclusion:

• $y = 0$
• $x = 250$
• $z = 550$
• Satellites serving C = 725 (uniquely determined)

Hence, the statement "the number of satellites serving C cannot be uniquely determined" is false.

Correct Answer: (A)