Question

At what integer value will the values of y=3(2)x
surpass the values of y=5x2+3
?(1 point)
x =

Answer 1

To determine the integer value of \( x \) at which the equation \( y = 3(2^x) \) surpasses \( y = 5x^2 + 3 \), we need to solve the inequality:

\[
3(2^x) > 5x^2 + 3.
\]

To find the point where this inequality holds, we can evaluate both sides for different integer values of \( x \).

1. **For \( x = 0 \)**:
- \( y = 3(2^0) = 3(1) = 3 \)
- \( y = 5(0^2) + 3 = 3 \)
- Result: \( 3 = 3 \) (not surpassed).

2. **For \( x = 1 \)**:
- \( y = 3(2^1) = 3(2) = 6 \)
- \( y = 5(1^2) + 3 = 5(1) + 3 = 8 \)
- Result: \( 6 < 8 \).

3. **For \( x = 2 \)**:
- \( y = 3(2^2) = 3(4) = 12 \)
- \( y = 5(2^2) + 3 = 5(4) + 3 = 20 + 3 = 23 \)
- Result: \( 12 < 23 \).

4. **For \( x = 3 \)**:
- \( y = 3(2^3) = 3(8) = 24 \)
- \( y = 5(3^2) + 3 = 5(9) + 3 = 45 + 3 = 48 \)
- Result: \( 24 < 48 \).

5. **For \( x = 4 \)**:
- \( y = 3(2^4) = 3(16) = 48 \)
- \( y = 5(4^2) + 3 = 5(16) + 3 = 80 + 3 = 83 \)
- Result: \( 48 < 83 \).

6. **For \( x = 5 \)**:
- \( y = 3(2^5) = 3(32) = 96 \)
- \( y = 5(5^2) + 3 = 5(25) + 3 = 125 + 3 = 128 \)
- Result: \( 96 < 128 \).

7. **For \( x = 6 \)**:
- \( y = 3(2^6) = 3(64) = 192 \)
- \( y = 5(6^2) + 3 = 5(36) + 3 = 180 + 3 = 183 \)
- Result: \( 192 > 183 \).

Thus, the values of \( y = 3(2^x) \) surpass \( y = 5x^2 + 3 \) for the first time at \( x = 6 \).

Therefore, the answer is:

\[
\boxed{6}
\]