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Chapter 7: Problem 69

Tani throws a ball up into the air so that its height \(h\) (in feet) above the ground \(t\) seconds after it is thrown is given by the formula \(h=5+80 t-16 t^{2} .\) Find the height of the ball after 1 second, 2 seconds, and 2.5 seconds.

Short Answer

Expert verified
69 feet, 101 feet, 105 feet

Step by step solution

01

Understanding the formula

The formula provided for the height of the ball is given by \( h = 5 + 80t - 16t^2 \). In this formula, \( h \) represents the height in feet and \( t \) represents the time in seconds.
02

Finding the height at 1 second

Substitute \( t = 1 \) into the formula\( h = 5 + 80t - 16t^2 \) and solve for \( h \):\[h = 5 + 80(1) - 16(1)^2 \] \[h = 5 + 80 - 16 \] \[h = 69 \] So, the height of the ball after 1 second is 69 feet.
03

Finding the height at 2 seconds

Substitute \( t = 2 \) into the formula\( h = 5 + 80t - 16t^2 \) and solve for \( h \):\[h = 5 + 80(2) - 16(2)^2 \] \[h = 5 + 160 - 64 \] \[h = 101 \] So, the height of the ball after 2 seconds is 101 feet.
04

Finding the height at 2.5 seconds

Substitute\( t = 2.5 \) into the formula\( h = 5 + 80t - 16t^2 \) and solve for\( h \):\[h = 5 + 80(2.5) - 16(2.5)^2 \] \[h = 5 + 200 - 100 \] \[h = 105 \] So, the height of the ball after 2.5 seconds is 105 feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parabolic motion
When Tani throws a ball up into the air, its movement follows a particular path called parabolic motion. This is typical for objects launched into the air, where gravity plays a crucial role in shaping their trajectory.
When you look at the height formula given: \(h = 5 + 80t - 16t^2\), you can see it forms a parabola because of the squared term \( -16t^2\). This squared component causes the ball to follow a curved path up and then back down.
Main points to remember about parabolic motion:
  • The highest point of the parabola is the peak height that the ball reaches.
  • Gravity constantly pulls the ball downwards, which is why the squared term has a negative coefficient.
  • The linear term (80t) pushes the ball up while the constant term (5) gives the initial height above the ground.
algebraic substitution
Algebraic substitution is a technique used to find the value of a variable by replacing it with another known value.
In this exercise to find the height of the ball after certain times, we substituted specific values of time (\(t = 1, 2, \text{and} 2.5\)) into the height formula. To find the height after 1 second, we substitute t = 1 into the formula:
begin{-align} \h = 5 + 80(1) - 16(1)^2\
This gives us the height of 69 feet.
By repeating similar steps for other values of t like 2 and 2.5 seconds, we can determine their respective heights.
  • Substitute the given value of time into the formula.
  • Perform arithmetic operations step by step to solve for height.
  • Record the result accurately.
height formula
The height formula used in this exercise is \(h = 5 + 80t - 16t^2\). This quadratic equation helps in calculating the height of the ball at any point in time after it has been thrown.
  • The constant term, 5, represents the initial height of the ball when Tani throws it.
  • The 80t term indicates how the height increases initially as time progresses.
  • The \( -16t^2 \) term accounts for the effect of gravity pulling the ball back down after it reaches a certain height.
It's a powerful tool to analyze and predict the motion of objects under the influence of gravity. By plugging in different values of t (time), you can find out how high the ball is at any given second after it leaves Tani's hand. The formula encapsulates the whole motion of the object in a single, easy-to-manipulate equation.

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Most popular questions from this chapter

perform the indicated operation and simplify. $$\frac{x}{3}-\frac{x}{4}$$

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

Convert each number into standard notation. $$3.5286 \times 10^{5}$$

Do your computations using scientific notation. Atomic measurements or other very small distances are often measured in units called angstroms. One angstrom, which is written \(1 \AA\), is equal to \(0.00000001 \mathrm{cm} .\) Write this number in scientific notation.

Do your computation using scientific notation. $$\frac{(0.0036)(0.005)}{(0.01)(0.06)}$$
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