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Answer :
To find the average velocity of the object between [tex]\(x = 1\)[/tex] and [tex]\(x = 1.8\)[/tex] seconds, we will follow these steps:
1. Understand the Function [tex]\( h(x) \)[/tex]:
- The height of the object is given by the function [tex]\( h(x) = -16x^2 + 149x + 186 \)[/tex].
- Here, [tex]\(x\)[/tex] represents time in seconds, and [tex]\(h(x)\)[/tex] is the height in feet.
2. Calculate the Heights at [tex]\(x = 1\)[/tex] and [tex]\(x = 1.8\)[/tex]:
- First, find the height at [tex]\(x = 1\)[/tex] using the function:
[tex]\[
h(1) = -16(1)^2 + 149(1) + 186 = 319 \, \text{feet}
\][/tex]
- Next, find the height at [tex]\(x = 1.8\)[/tex]:
[tex]\[
h(1.8) = -16(1.8)^2 + 149(1.8) + 186 = 402.36 \, \text{feet}
\][/tex]
3. Calculate the Average Velocity:
- The average velocity is given by the formula:
[tex]\[
\text{Average velocity} = \frac{h(x_2) - h(x_1)}{x_2 - x_1}
\][/tex]
- Here, [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = 1.8\)[/tex], and we just calculated [tex]\(h(1) = 319\)[/tex] and [tex]\(h(1.8) = 402.36\)[/tex].
- Substitute these values into the formula:
[tex]\[
\text{Average velocity} = \frac{402.36 - 319}{1.8 - 1} = \frac{83.36}{0.8} = 104.2 \, \text{feet per second}
\][/tex]
Therefore, the average velocity of the object between [tex]\(x = 1\)[/tex] second and [tex]\(x = 1.8\)[/tex] seconds is approximately [tex]\(104.2\)[/tex] feet per second.
1. Understand the Function [tex]\( h(x) \)[/tex]:
- The height of the object is given by the function [tex]\( h(x) = -16x^2 + 149x + 186 \)[/tex].
- Here, [tex]\(x\)[/tex] represents time in seconds, and [tex]\(h(x)\)[/tex] is the height in feet.
2. Calculate the Heights at [tex]\(x = 1\)[/tex] and [tex]\(x = 1.8\)[/tex]:
- First, find the height at [tex]\(x = 1\)[/tex] using the function:
[tex]\[
h(1) = -16(1)^2 + 149(1) + 186 = 319 \, \text{feet}
\][/tex]
- Next, find the height at [tex]\(x = 1.8\)[/tex]:
[tex]\[
h(1.8) = -16(1.8)^2 + 149(1.8) + 186 = 402.36 \, \text{feet}
\][/tex]
3. Calculate the Average Velocity:
- The average velocity is given by the formula:
[tex]\[
\text{Average velocity} = \frac{h(x_2) - h(x_1)}{x_2 - x_1}
\][/tex]
- Here, [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = 1.8\)[/tex], and we just calculated [tex]\(h(1) = 319\)[/tex] and [tex]\(h(1.8) = 402.36\)[/tex].
- Substitute these values into the formula:
[tex]\[
\text{Average velocity} = \frac{402.36 - 319}{1.8 - 1} = \frac{83.36}{0.8} = 104.2 \, \text{feet per second}
\][/tex]
Therefore, the average velocity of the object between [tex]\(x = 1\)[/tex] second and [tex]\(x = 1.8\)[/tex] seconds is approximately [tex]\(104.2\)[/tex] feet per second.
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