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Jobisdone [24]
3 years ago
8

The coefficient of the third term in the expansion of the binomial (3x^2+2y^3)^4 is?

Mathematics
1 answer:
babunello [35]3 years ago
8 0
(3x² + 2y³)⁴
(3x² + 2y³)²(3x² + 2y³)²
(3x² + 2y³)(3x² + 2y³)(3x² + 2y³)(3x² + 2y³)
(3x²(3x² + 2y³) + 2y³(3x² + 2y³))(3x²(3x² + 2y³) + 2y³(3x² + 2y³))
(3x²(3x²) + 3x²(2y³) + 2y³(3x²) + 2y³(2y³))(3x²(3x²) + 3x²(2y³) + 2y³(3x²) + 2y³(2y³))
(6x⁴ + 6x²y³ + 6x²y³ + 4y⁶)(6x⁴ + 6x²y³ + 6x²y³ + 4y⁶)
(6x⁴ + 12x²y³ + 4y⁶)(6x⁴ + 12x²y³ + 4y⁶)
6x⁴(6x⁴ + 12x²y³ + 4y⁶) + 12x²y³(6x⁴ + 12x²y³ + 4y⁶) + 4y⁶(6x⁴ + 12x²y³ + 4y⁶)
6x⁴(6x⁴) + 6x⁴(12x²y³) + 6x⁴(4y⁶) + 12x²y³(6x⁴) + 12x²y³(12x²y³) + 12x²y³(4y⁶) + 4y⁵(6x⁴) + 4y⁶(12x²y³) + 4y⁶(4y⁶)
24x⁸ + 72x⁶y³ + 24x⁴y⁶ + 72x⁶y³ + 144x⁴y⁶ + 48x²y⁹ + 24x⁴y⁵ + 48x²y⁹ + 16y¹²
24x⁸ + 72x⁶y³ + 72x⁶y³ + 24x⁴y⁶ + 144x⁴y⁶ + 48x²y⁹ + 48x²y⁹ + 24x⁴y⁵ + 16y¹²
24x⁸ + 144x⁶y³ + 168x⁴y⁶ + 96x²y⁹ + 24x⁴y⁵ + 16y¹²

The coefficient of the third term in the expansion of the binomial is 6.
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(PART-A): s-intercept in this case represents the y-intercept of the graph, whereas the t-intercept in this case represents the x-intercept of the graph.

Always remember that y-intercept represents the value of y where the curve (or line) crosses the y-axis, and x-intercept represents the value of x where the curve (or line) crosses the x-axis.

In the graph attached with the question, at point (0,6) the curve crosses the y-axis. Therefore, the <em>y-intercept (s-intercept) in this case is 6</em>. It means that in year 2000 (t=0), 6 hundred shirts were sold.

However, there is no point on the graph where the curve crosses the x-axis, meaning there is <em>no x-intercept (t-intercept)</em>. It means that there is <em>not</em> a single year when the number of shirts sold is 0.


(PART B): As you can see in the graph (attached with the question) that the <em>f(t)</em> is increasing (exponentially) with the increase in <em>t. </em>Therefore, we can safely say that as t increases without bound, the f(t) increases (also). In this context, it means that the sale of shirts increases as the years go by. Hence, the correct blank is "increases."


(PART C): The average rate of change is actually the slope. To find the slope, we can use the following formula:

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slope = \frac{22-12.5}{7-5} = 4.75

Hence, the average rate of change for the function between t=5 and t=7 is 4.75 (answer).


(PART D): This part is bit tricky. Therefore, read the explanation carefully!

You can see in the graph that number of Shirts sold is in <em>hundreds. </em>It means that 1 unit (of y-axis in a graph) represents 100 shirts. Therefore, as in the question it is mentioned that there are 1000 T-shirts sold, it will become 10 units (since 10*100 = 1000). So, the function g(t) will become the following:

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