$$ \sqrt{a-2} $$=2$$ \sqrt{b+1} $$   equation 1                                                                                                                                                      $$ 3^{\left(c-4\right)} $$+1=$$ a\cdot b $$          equation 2                                                                                                                                                       $$ \log_{b}\left(d\right) $$$$ -1=\log_{b}\left(d-16\right) $$  equation 3                                                                                                                               $$ \log\left(a-4\right) $$=1                     equation 4                                                                                                                                          $$ \int_{aa}^{d}\!b\,\mathrm{dt-} $$1900($$ a\cdot b\cdot c\cdot d) $$=? what will be the answer ? is there any possibility that answer would be from this options (32,16,2,240,224,14)

Let's solve the system of equations and then determine the value of the final expression. The steps to solve the equations will be the same as before, but we'll calculate the final expression accordingly.

### Step 1: Solve Equation 4
\[
\log(a - 4) = 1
\]
This implies:
\[
a - 4 = 10^1 = 10
\]
\[
a = 14
\]

### Step 2: Solve Equation 1
\[
\sqrt{a - 2} = 2\sqrt{b + 1}
\]
Substitute $ a = 14 $ into the equation:
\[
\sqrt{14 - 2} = 2\sqrt{b + 1}
\]
\[
\sqrt{12} = 2\sqrt{b + 1}
\]
Square both sides:
\[
12 = 4(b + 1)
\]
\[
b + 1 = 3
\]
\[
b = 2
\]

### Step 3: Solve Equation 2
\[
3^{(c - 4)} + 1 = a \cdot b
\]
Substitute $ a = 14 $ and $ b = 2 $:
\[
3^{(c - 4)} + 1 = 14 \times 2
\]
\[
3^{(c - 4)} + 1 = 28
\]
\[
3^{(c - 4)} = 27
\]
\[
c - 4 = 3
\]
\[
c = 7
\]

### Step 4: Solve Equation 3
\[
\log_b(d) - 1 = \log_b(d - 16)
\]
Since $ b = 2 $, we have:
\[
\log_2(d) - 1 = \log_2(d - 16)
\]
Add 1 to both sides:
\[
\log_2(d) = \log_2(2 \cdot (d - 16))
\]
This implies:
\[
d = 2(d - 16)
\]
\[
d = 2d - 32
\]
\[
32 = d
\]

### Step 5: Calculate the Integral and the Final Expression
We need to calculate the integral from $ aa $ to $ d $ of $ b $ with respect to $ t $, where $ aa = 14^2 = 196 $:
\[
\int_{196}^{32} b \, dt - 1900 \times (a \cdot b \cdot c \cdot d)
\]
Since $ b = 2 $, we have:
\[
\int_{196}^{32} 2 \, dt = 2 \times (32 - 196) = 2 \times (-164) = -328
\]

Now, let's calculate the product $ a \cdot b \cdot c \cdot d $:
\[
a \cdot b \cdot c \cdot d = 14 \times 2 \times 7 \times 32 = 6272
\]

Then, calculate the final expression:
\[
-328 - 1900 \times 6272
\]

Let's simplify this calculation:
\[
1900 \times 6272 = 11916800
\]
\[
-328 - 11916800 = -11917128
\]

This result is a large negative number. Given the options provided (32, 16, 2, 240, 224, 14), none of them directly match this outcome. 

Therefore, the final expression does **not** match any of the provided options.

Would you like to explore further calculations or analyze it differently?

---
**5 relative questions**:
1. How would the integral change if we swapped the limits of integration?
2. Can the logarithmic bases be altered to find alternative solutions?
3. What are the effects of modifying the value of $ m $ in the final expression?
4. What happens if we adjust the value of $ b $ to a different integer?
5. Is there a simplification approach for the large multiplication step?

**Tip**: When working with large calculations, check if factors or modular arithmetic can simplify the process.

Learn how to solve a system of equations involving square roots, exponentials, logarithms, and integrals step-by-step. This problem covers advanced algebraic concepts, including quadratic equations, exponential properties, logarithmic identities, and integral calculations. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation