Can someone help me answer this?
If a < 5 the define integral [a, 4] 2.4e^(1.4x)dx = 44
Find the value a
Define integral = integral sign
a = lower limit
4 = upper limit
Sure, I can help you with that!
To solve this equation, you need to apply the fundamental theorem of calculus and evaluate the definite integral. Here's how you can do it step by step:
1. First, find the antiderivative of the function inside the integral. In this case, the function is 2.4e^(1.4x), and its antiderivative is (2.4/1.4)e^(1.4x).
2. Now, using the lower and upper limits of integration, substitute them into the antiderivative equation. For the lower limit a, you would have (2.4/1.4)e^(1.4a), and for the upper limit 4, you would have (2.4/1.4)e^(1.4*4).
3. Subtract the result from the lower limit from the result from the upper limit to get the value of the definite integral. In this case, you would have (2.4/1.4)e^(1.4*4) - (2.4/1.4)e^(1.4a) = 44.
4. Simplify the equation by dividing both sides by (2.4/1.4), which is approximately 1.7142857. The equation becomes e^(1.4*4) - e^(1.4a) ≈ 25.7142857.
5. Next, we need to isolate the variable "a" in the equation. To do that, bring e^(1.4a) to the other side of the equation by subtracting it from both sides. You will have e^(1.4*4) ≈ 25.7142857 + e^(1.4a).
6. Take the natural logarithm of both sides of the equation to cancel out the exponential term. This will give you ln(e^(1.4*4)) ≈ ln(25.7142857 + e^(1.4a)).
7. Simplify the equation to 1.4*4 ≈ ln(25.7142857 + e^(1.4a)).
8. Evaluate the left side of the equation to get 5.6 ≈ ln(25.7142857 + e^(1.4a)).
9. Now, isolate the variable "a" by subtracting 25.7142857 from both sides of the equation. You'll have -20.1142857 ≈ ln(e^(1.4a)).
10. Convert the logarithmic equation back to an exponential equation using the inverse of the natural logarithm, which is e^x. You will have e^(-20.1142857) ≈ e^(1.4a).
11. Cancel out e^(1.4a) from both sides of the equation by applying the natural logarithm to both sides. This will give you -20.1142857 ≈ 1.4a.
12. Finally, solve for "a" by dividing both sides of the equation by 1.4. You will get approximately a ≈ -14.367346.
So, the value of "a" that satisfies the equation is approximately -14.367346.