Questions: Homework: HW Section 2.1b Question 4, 2.1.92-LS Part 3 of 4 Points: 0.5 of 1 The percentage s of seats in the House of Representatives won by Republicans and the percentage v of votes cast for Republicans (when expressed as decimal fractions) are related by the following equation. 25v-16s=4.3 0<s<1,0.172<v<0.812 (A) Express v as a function of s and find the percentage of votes required for Republicans to win 51% of the seats. v(s)=.64s+.172 For Republicans to win 51% of the seats, they need to receive 49.8% of the votes. (B) Express s as a function of v. s(v)=

Homework: HW Section 2.1b Question 4, 2.1.92-LS Part 3 of 4 Points: 0.5 of 1 The percentage s of seats in the House of Representatives won by Republicans and the percentage v of votes cast for Republicans (when expressed as decimal fractions) are related by the following equation. 25v-16s=4.3 0<s<1,0.172<v<0.812 (A) Express v as a function of s and find the percentage of votes required for Republicans to win 51% of the seats. v(s)=.64s+.172 For Republicans to win 51% of the seats, they need to receive 49.8% of the votes. (B) Express s as a function of v. s(v)=
Transcript text: Homework: HW Section 2.1b Question 4, 2.1.92-LS Part 3 of 4 Points: 0.5 of 1 The percentage $s$ of seats in the House of Representatives won by Republicans and the percentage $v$ of votes cast for Republicans (when expressed as decimal fractions) are related by the following equation. \[ 25 v-16 s=4.3 \quad 0
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Solution

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Solution Steps

To solve the given problem, we need to manipulate the given equation to express \( v \) as a function of \( s \) and vice versa. Then, we will use these functions to find the required percentages.

Part (A)
  1. Start with the given equation: \( 25v - 16s = 4.3 \).
  2. Solve for \( v \) in terms of \( s \).
  3. Substitute \( s = 0.51 \) to find the percentage of votes required.
Part (B)
  1. Use the same equation: \( 25v - 16s = 4.3 \).
  2. Solve for \( s \) in terms of \( v \).
  3. Substitute \( v = 0.51 \) to find the percentage of seats won.
Step 1: Express \( v \) as a Function of \( s \)

Starting with the equation:

\[ 25v - 16s = 4.3 \]

we can solve for \( v \):

\[ v(s) = 0.64s + 0.172 \]

Step 2: Calculate Votes Required for 51% of Seats

To find the percentage of votes required for Republicans to win \( 51\% \) of the seats, we substitute \( s = 0.51 \) into the function \( v(s) \):

\[ v(0.51) = 0.64(0.51) + 0.172 = 0.4984 \]

Thus, the percentage of votes required is:

\[ \text{Votes Required} = 49.84\% \]

Step 3: Express \( s \) as a Function of \( v \)

Next, we rearrange the original equation to express \( s \) in terms of \( v \):

\[ s(v) = \frac{25v - 4.3}{16} \]

Step 4: Calculate Seats Won for 51% of Votes

To find the percentage of seats won if Republicans receive \( 51\% \) of the votes, we substitute \( v = 0.51 \) into the function \( s(v) \):

\[ s(0.51) = \frac{25(0.51) - 4.3}{16} = 0.528125 \]

Thus, the percentage of seats won is:

\[ \text{Seats Won} = 52.8125\% \]

Final Answer

The results are:

  • The percentage of votes required for Republicans to win \( 51\% \) of the seats is \( 49.84\% \).
  • The percentage of seats won if Republicans receive \( 51\% \) of the votes is \( 52.8125\% \).

\[ \boxed{\text{Votes Required} = 49.8\%, \text{Seats Won} = 52.8\%} \]

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