Math Problem Statement

a2\sqrt{a-2}=2b+1\sqrt{b+1} equation 1 3(c4)3^{\left(c-4\right)}+1=aba\cdot b equation 2 \log_{b}\left(d\right) $$$$ -1=\log_{b}\left(d-16\right) equation 3 log(a4)\log\left(a-4\right)=1 equation 4 aad ⁣bdt\int_{aa}^{d}\!b\,\mathrm{dt-}1900(abcd)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)

Solution

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(a4)=1\log(a - 4) = 1 This implies: a4=101=10a - 4 = 10^1 = 10 a=14a = 14

Step 2: Solve Equation 1

a2=2b+1\sqrt{a - 2} = 2\sqrt{b + 1} Substitute a=14a = 14 into the equation: 142=2b+1\sqrt{14 - 2} = 2\sqrt{b + 1} 12=2b+1\sqrt{12} = 2\sqrt{b + 1} Square both sides: 12=4(b+1)12 = 4(b + 1) b+1=3b + 1 = 3 b=2b = 2

Step 3: Solve Equation 2

3(c4)+1=ab3^{(c - 4)} + 1 = a \cdot b Substitute a=14a = 14 and b=2b = 2: 3(c4)+1=14×23^{(c - 4)} + 1 = 14 \times 2 3(c4)+1=283^{(c - 4)} + 1 = 28 3(c4)=273^{(c - 4)} = 27 c4=3c - 4 = 3 c=7c = 7

Step 4: Solve Equation 3

logb(d)1=logb(d16)\log_b(d) - 1 = \log_b(d - 16) Since b=2b = 2, we have: log2(d)1=log2(d16)\log_2(d) - 1 = \log_2(d - 16) Add 1 to both sides: log2(d)=log2(2(d16))\log_2(d) = \log_2(2 \cdot (d - 16)) This implies: d=2(d16)d = 2(d - 16) d=2d32d = 2d - 32 32=d32 = d

Step 5: Calculate the Integral and the Final Expression

We need to calculate the integral from aaaa to dd of bb with respect to tt, where aa=142=196aa = 14^2 = 196: 19632bdt1900×(abcd)\int_{196}^{32} b \, dt - 1900 \times (a \cdot b \cdot c \cdot d) Since b=2b = 2, we have: 196322dt=2×(32196)=2×(164)=328\int_{196}^{32} 2 \, dt = 2 \times (32 - 196) = 2 \times (-164) = -328

Now, let's calculate the product abcda \cdot b \cdot c \cdot d: abcd=14×2×7×32=6272a \cdot b \cdot c \cdot d = 14 \times 2 \times 7 \times 32 = 6272

Then, calculate the final expression: 3281900×6272-328 - 1900 \times 6272

Let's simplify this calculation: 1900×6272=119168001900 \times 6272 = 11916800 32811916800=11917128-328 - 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 mm in the final expression?
  4. What happens if we adjust the value of bb 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.

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Math Problem Analysis

Mathematical Concepts

Square roots
Exponentials
Logarithms
Integrals

Formulas

Quadratic formula
Exponential properties
Logarithmic identities
Integral calculation

Theorems

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Suitable Grade Level

Advanced High School