Question
12.
A rock is dropped from a bridge 320 feet above a river. The pathway that the rock takes can be modeled by the equation below. Approximately how long will it take the rock to reach the river?
h(t) = −16t2+320
(1 point)
Responses
2.5 seconds
2.5 seconds
3.5 seconds
3.5 seconds
3.8 seconds
3.8 seconds
4.5 seconds
4.5 seconds
Question 2
13.
Solve the equation. Simplify all irrational solutions.
5x2 +14x = x + 6
(1 point)
Responses
x={−3 , 25}
x={−3 , 25}
x={−3, 5}
x= {−3, 5}
x={15, 5}
x={15, 5}
x={23, 3}
x={23, 3}
Question 3
14.
Solve the equation. Simplify all irrational solutions.
x2 − 6x − 23 = 0
(1 point)
Responses
x = {4±32–√}
x = {4±32–√}
x = {3±42–√}
x = {3±42–√}
x = {±32−−√}
x = {±32−−√}
x = {−3±42–√}
x = {−3±42–√}
Question 4
15.
What is the value of c that completes the square trinomial to give the two solutions {17, -1}?
x2 − 16x − 13 = 4
(1 point)
Responses
17
17
81
81
64
64
8
8
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A rock is dropped from a bridge 320 feet above a river. The pathway that the rock takes can be modeled by the equation below. Approximately how long will it take the rock to reach the river?
h(t) = −16t2+320
(1 point)
Responses
2.5 seconds
2.5 seconds
3.5 seconds
3.5 seconds
3.8 seconds
3.8 seconds
4.5 seconds
4.5 seconds
Question 2
13.
Solve the equation. Simplify all irrational solutions.
5x2 +14x = x + 6
(1 point)
Responses
x={−3 , 25}
x={−3 , 25}
x={−3, 5}
x= {−3, 5}
x={15, 5}
x={15, 5}
x={23, 3}
x={23, 3}
Question 3
14.
Solve the equation. Simplify all irrational solutions.
x2 − 6x − 23 = 0
(1 point)
Responses
x = {4±32–√}
x = {4±32–√}
x = {3±42–√}
x = {3±42–√}
x = {±32−−√}
x = {±32−−√}
x = {−3±42–√}
x = {−3±42–√}
Question 4
15.
What is the value of c that completes the square trinomial to give the two solutions {17, -1}?
x2 − 16x − 13 = 4
(1 point)
Responses
17
17
81
81
64
64
8
8
Skip to navigation
Answers
GPT-4o mini
Let's solve each question step by step.
**Question 1: Time for the rock to reach the river**
The height of the rock as a function of time is given by:
\[ h(t) = -16t^2 + 320 \]
We want to find the time \( t \) when the rock reaches the river, which occurs when \( h(t) = 0 \):
\[ -16t^2 + 320 = 0 \]
To solve for \( t \):
1. Rearranging gives:
\[ -16t^2 = -320 \]
2. Dividing both sides by -16:
\[ t^2 = 20 \]
3. Taking the square root:
\[ t = \sqrt{20} \approx 4.47 \]
So, the time it takes approximately is 4.5 seconds.
**Answer:** **4.5 seconds**
---
**Question 2: Solve the equation \( 5x^2 + 14x = x + 6 \)**
1. Rearranging the equation gives:
\[ 5x^2 + 14x - x - 6 = 0 \]
\[ 5x^2 + 13x - 6 = 0 \]
2. We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- \( a = 5 \), \( b = 13 \), \( c = -6 \)
- Discriminant:
\[ b^2 - 4ac = 13^2 - 4 \cdot 5 \cdot (-6) = 169 + 120 = 289 \]
- Roots:
\[ x = \frac{-13 \pm \sqrt{289}}{10} \]
\[ x = \frac{-13 \pm 17}{10} \]
- Thus,
\[ x_1 = \frac{4}{10} = 0.4 \]
\[ x_2 = \frac{-30}{10} = -3 \]
From the answer choices, we find the solutions as:
\[ x = \{-3, 0.4\} \]
Since \( 0.4 \) isn’t listed in the options, let’s check the given options closely:
The closest option that matches the given solutions might require re-evaluation.
3. None of the provided responses seem accurate so the expected answer might be **x = -3, 0.4** or something else, but it's not listed correctly.
---
**Question 3: Solve the equation \( x^2 - 6x - 23 = 0 \)**
Again, using the quadratic formula:
1. Here \( a = 1 \), \( b = -6 \), \( c = -23 \)
2. The discriminant:
\[ b^2 - 4ac = (-6)^2 - 4(1)(-23) = 36 + 92 = 128 \]
3. Roots:
\[ x = \frac{6 \pm \sqrt{128}}{2 \cdot 1} \]
\[ x = \frac{6 \pm 8\sqrt{2}}{2} \]
\[ x = 3 \pm 4\sqrt{2} \]
Thus the correct answer is:
**x = {3 ± 4√2}**
---
**Question 4: Completing the square**
Given:
\[ x^2 - 16x - 13 = 4 \]
To complete the square:
1. Rearranging:
\[ x^2 - 16x = 17 \]
2. Completing the square involves taking half of -16, squaring it:
- Half of -16 is -8, and \((-8)^2 = 64\)
3. Thus,
\[ (x - 8)^2 = 17 + 64 \]
\[ (x - 8)^2 = 81 \]
Thus, the number that completes the square is:
**64**
---
Final Answers Recap:
1. **4.5 seconds**
2. The responses might require clarification since no option matches correctly.
3. **x = {3 ± 4√2}**
4. **64**
**Question 1: Time for the rock to reach the river**
The height of the rock as a function of time is given by:
\[ h(t) = -16t^2 + 320 \]
We want to find the time \( t \) when the rock reaches the river, which occurs when \( h(t) = 0 \):
\[ -16t^2 + 320 = 0 \]
To solve for \( t \):
1. Rearranging gives:
\[ -16t^2 = -320 \]
2. Dividing both sides by -16:
\[ t^2 = 20 \]
3. Taking the square root:
\[ t = \sqrt{20} \approx 4.47 \]
So, the time it takes approximately is 4.5 seconds.
**Answer:** **4.5 seconds**
---
**Question 2: Solve the equation \( 5x^2 + 14x = x + 6 \)**
1. Rearranging the equation gives:
\[ 5x^2 + 14x - x - 6 = 0 \]
\[ 5x^2 + 13x - 6 = 0 \]
2. We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- \( a = 5 \), \( b = 13 \), \( c = -6 \)
- Discriminant:
\[ b^2 - 4ac = 13^2 - 4 \cdot 5 \cdot (-6) = 169 + 120 = 289 \]
- Roots:
\[ x = \frac{-13 \pm \sqrt{289}}{10} \]
\[ x = \frac{-13 \pm 17}{10} \]
- Thus,
\[ x_1 = \frac{4}{10} = 0.4 \]
\[ x_2 = \frac{-30}{10} = -3 \]
From the answer choices, we find the solutions as:
\[ x = \{-3, 0.4\} \]
Since \( 0.4 \) isn’t listed in the options, let’s check the given options closely:
The closest option that matches the given solutions might require re-evaluation.
3. None of the provided responses seem accurate so the expected answer might be **x = -3, 0.4** or something else, but it's not listed correctly.
---
**Question 3: Solve the equation \( x^2 - 6x - 23 = 0 \)**
Again, using the quadratic formula:
1. Here \( a = 1 \), \( b = -6 \), \( c = -23 \)
2. The discriminant:
\[ b^2 - 4ac = (-6)^2 - 4(1)(-23) = 36 + 92 = 128 \]
3. Roots:
\[ x = \frac{6 \pm \sqrt{128}}{2 \cdot 1} \]
\[ x = \frac{6 \pm 8\sqrt{2}}{2} \]
\[ x = 3 \pm 4\sqrt{2} \]
Thus the correct answer is:
**x = {3 ± 4√2}**
---
**Question 4: Completing the square**
Given:
\[ x^2 - 16x - 13 = 4 \]
To complete the square:
1. Rearranging:
\[ x^2 - 16x = 17 \]
2. Completing the square involves taking half of -16, squaring it:
- Half of -16 is -8, and \((-8)^2 = 64\)
3. Thus,
\[ (x - 8)^2 = 17 + 64 \]
\[ (x - 8)^2 = 81 \]
Thus, the number that completes the square is:
**64**
---
Final Answers Recap:
1. **4.5 seconds**
2. The responses might require clarification since no option matches correctly.
3. **x = {3 ± 4√2}**
4. **64**