Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Let's break down the solution into several steps:
1. Height of the spray 2 feet away from the sprinkler head:
We use the given formula [tex]\( h(x) = 160x - 16x^2 \)[/tex] to find the height when [tex]\( x = 2 \)[/tex].
[tex]\[ h(2) = 160 \times 2 - 16 \times 2^2 \][/tex]
Simplifying inside the equation:
[tex]\[ h(2) = 320 - 16 \times 4 \][/tex]
[tex]\[ h(2) = 320 - 64 \][/tex]
[tex]\[ h(2) = 256 \][/tex]
Therefore, after 2 feet, the height of the spray is [tex]\( \boxed{256} \)[/tex] inches.
2. Distance along the ground where the spray reaches maximum height:
The height function [tex]\( h(x) = 160x - 16x^2 \)[/tex] represents a parabola that opens downwards (since the coefficient of [tex]\( x^2 \)[/tex] is negative). The maximum height occurs at the vertex of the parabola.
For a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -16 \)[/tex] and [tex]\( b = 160 \)[/tex]. Substituting these values in:
[tex]\[ x = -\frac{160}{2 \times -16} \][/tex]
[tex]\[ x = -\frac{160}{-32} \][/tex]
[tex]\[ x = 5 \][/tex]
So, the spray reaches its maximum height at [tex]\( \boxed{5} \)[/tex] feet away from the sprinkler head.
3. Maximum height of the water spray:
We already found the x-coordinate where the maximum height occurs (5 feet away). To find the maximum height:
[tex]\[ h(5) = 160 \times 5 - 16 \times 5^2 \][/tex]
Simplifying inside the equation:
[tex]\[ h(5) = 800 - 16 \times 25 \][/tex]
[tex]\[ h(5) = 800 - 400 \][/tex]
[tex]\[ h(5) = 400 \][/tex]
Therefore, the maximum height of the water spray is [tex]\( \boxed{400} \)[/tex] inches.
4. Distance away from the sprinkler head where the water hits the ground again:
The water hits the ground whenever the height [tex]\( h(x) \)[/tex] is zero. Therefore, we solve the equation:
[tex]\[ 0 = 160x - 16x^2 \][/tex]
Factoring out the common terms:
[tex]\[ 0 = x (160 - 16x) \][/tex]
This gives us two solutions:
[tex]\[ x = 0 \quad \text{or} \quad 160 - 16x = 0 \][/tex]
Solving [tex]\( 160 - 16x = 0 \)[/tex]:
[tex]\[ 160 = 16x \][/tex]
[tex]\[ x = 10 \][/tex]
Therefore, the water hits the ground again at [tex]\( \boxed{10} \)[/tex] feet away from the sprinkler head.
1. Height of the spray 2 feet away from the sprinkler head:
We use the given formula [tex]\( h(x) = 160x - 16x^2 \)[/tex] to find the height when [tex]\( x = 2 \)[/tex].
[tex]\[ h(2) = 160 \times 2 - 16 \times 2^2 \][/tex]
Simplifying inside the equation:
[tex]\[ h(2) = 320 - 16 \times 4 \][/tex]
[tex]\[ h(2) = 320 - 64 \][/tex]
[tex]\[ h(2) = 256 \][/tex]
Therefore, after 2 feet, the height of the spray is [tex]\( \boxed{256} \)[/tex] inches.
2. Distance along the ground where the spray reaches maximum height:
The height function [tex]\( h(x) = 160x - 16x^2 \)[/tex] represents a parabola that opens downwards (since the coefficient of [tex]\( x^2 \)[/tex] is negative). The maximum height occurs at the vertex of the parabola.
For a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -16 \)[/tex] and [tex]\( b = 160 \)[/tex]. Substituting these values in:
[tex]\[ x = -\frac{160}{2 \times -16} \][/tex]
[tex]\[ x = -\frac{160}{-32} \][/tex]
[tex]\[ x = 5 \][/tex]
So, the spray reaches its maximum height at [tex]\( \boxed{5} \)[/tex] feet away from the sprinkler head.
3. Maximum height of the water spray:
We already found the x-coordinate where the maximum height occurs (5 feet away). To find the maximum height:
[tex]\[ h(5) = 160 \times 5 - 16 \times 5^2 \][/tex]
Simplifying inside the equation:
[tex]\[ h(5) = 800 - 16 \times 25 \][/tex]
[tex]\[ h(5) = 800 - 400 \][/tex]
[tex]\[ h(5) = 400 \][/tex]
Therefore, the maximum height of the water spray is [tex]\( \boxed{400} \)[/tex] inches.
4. Distance away from the sprinkler head where the water hits the ground again:
The water hits the ground whenever the height [tex]\( h(x) \)[/tex] is zero. Therefore, we solve the equation:
[tex]\[ 0 = 160x - 16x^2 \][/tex]
Factoring out the common terms:
[tex]\[ 0 = x (160 - 16x) \][/tex]
This gives us two solutions:
[tex]\[ x = 0 \quad \text{or} \quad 160 - 16x = 0 \][/tex]
Solving [tex]\( 160 - 16x = 0 \)[/tex]:
[tex]\[ 160 = 16x \][/tex]
[tex]\[ x = 10 \][/tex]
Therefore, the water hits the ground again at [tex]\( \boxed{10} \)[/tex] feet away from the sprinkler head.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
