Math / Algebra

Question7. The height above the ground of a bungee jumper is modelled by the quadratic function $h(t) = -5(t - 0.3)^2 + 110$ , where height, $h(t)$ , is in metres and time, $t$ , is in seconds.
a) When does the bungee jumper reach maximum height? Why is it a maximum?
b) What is the maximum height reached by the jumper?
c) Determine the height of the platform from which the bungee jumper jumps.

Studdy Solution

STEP 1

What is this asking? We're finding out when a bungee jumper reaches their highest point, how high that is, and how high the platform is! Watch out! Don't mix up time and height!
Also, make sure you understand what each part of the equation means.

STEP 2

1. Find the time of maximum height.
2. Calculate the maximum height.
3. Determine the platform height.

STEP 3

Alright, so we've got this awesome equation $h(t) = -5(t - 0.3)^2 + 110$ where $t$ is the time and $h(t)$ is how high the jumper is at that time.
We want to find the biggest value of $h(t)$ , the maximum height.

STEP 4

Notice that pesky negative sign in front of the 5?
That tells us the squared term will always be negative (or zero) since anything squared is positive (or zero).
Multiplying by $-5$ makes the whole term $-5(t - 0.3)^2$ negative (or zero).

STEP 5

Since we're subtracting that term from 110, to get the biggest possible value for $h(t)$ , we want to subtract the smallest possible amount.
The smallest $-5(t - 0.3)^2$ can be is zero, which happens when $t - 0.3 = 0$ .

STEP 6

Solve for $\mathbf{t}$ : $t - 0.3 = 0$ $t = 0.3$ So, the jumper reaches the maximum height at $t = \mathbf{0.3}$ seconds!
It's a maximum because any other value of $t$ would make $-5(t - 0.3)^2$ negative, and subtracting a negative number is the same as adding a positive number, which would make $h(t)$ smaller.

STEP 7

Now, let's plug our time of maximum height, $t = 0.3$ , back into our equation to find out just how high that is!

STEP 8

Substitute $\mathbf{t=0.3}$ into the equation: $h(0.3) = -5(0.3 - 0.3)^2 + 110$ $h(0.3) = -5(0)^2 + 110$ $h(0.3) = -5 \cdot 0 + 110$ $h(0.3) = 0 + 110$ $h(0.3) = 110$ The maximum height is $\mathbf{110}$ meters!

STEP 9

The platform height is just the jumper's height at the very beginning, when $t = 0$ .
Let's plug that into our equation!

STEP 10

Substitute $\mathbf{t=0}$ into the equation: $h(0) = -5(0 - 0.3)^2 + 110$ $h(0) = -5(-0.3)^2 + 110$ $h(0) = -5(0.09) + 110$ $h(0) = -0.45 + 110$ $h(0) = 109.55$ So, the platform is $\mathbf{109.55}$ meters high!

STEP 11

a) The bungee jumper reaches maximum height at $\mathbf{0.3}$ seconds. b) The maximum height reached is $\mathbf{110}$ meters. c) The platform height is $\mathbf{109.55}$ meters.

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Studdy Solution

STEP 1

What is this asking? We're finding out when a bungee jumper reaches their highest point, how high that is, and how high the platform is! Watch out! Don't mix up time and height!
Also, make sure you understand what each part of the equation means.

STEP 2

1. Find the time of maximum height.
2. Calculate the maximum height.
3. Determine the platform height.

STEP 3

Alright, so we've got this awesome equation $h(t) = -5(t - 0.3)^2 + 110$ where $t$ is the time and $h(t)$ is how high the jumper is at that time.
We want to find the biggest value of $h(t)$ , the maximum height.

STEP 4

Notice that pesky negative sign in front of the 5?
That tells us the squared term will always be negative (or zero) since anything squared is positive (or zero).
Multiplying by $-5$ makes the whole term $-5(t - 0.3)^2$ negative (or zero).

STEP 5

Since we're subtracting that term from 110, to get the biggest possible value for $h(t)$ , we want to subtract the smallest possible amount.
The smallest $-5(t - 0.3)^2$ can be is zero, which happens when $t - 0.3 = 0$ .

STEP 6

STEP 7

Now, let's plug our time of maximum height, $t = 0.3$ , back into our equation to find out just how high that is!

STEP 8

Substitute $\mathbf{t=0.3}$ into the equation: $h(0.3) = -5(0.3 - 0.3)^2 + 110$ $h(0.3) = -5(0)^2 + 110$ $h(0.3) = -5 \cdot 0 + 110$ $h(0.3) = 0 + 110$ $h(0.3) = 110$ The maximum height is $\mathbf{110}$ meters!

STEP 9

The platform height is just the jumper's height at the very beginning, when $t = 0$ .
Let's plug that into our equation!

STEP 10

Substitute $\mathbf{t=0}$ into the equation: $h(0) = -5(0 - 0.3)^2 + 110$ $h(0) = -5(-0.3)^2 + 110$ $h(0) = -5(0.09) + 110$ $h(0) = -0.45 + 110$ $h(0) = 109.55$ So, the platform is $\mathbf{109.55}$ meters high!

STEP 11

a) The bungee jumper reaches maximum height at $\mathbf{0.3}$ seconds. b) The maximum height reached is $\mathbf{110}$ meters. c) The platform height is $\mathbf{109.55}$ meters.

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Studdy Solution

STEP 1

What is this asking? We're finding out when a bungee jumper reaches their highest point, how high that is, and how high the platform is! Watch out! Don't mix up time and height!Also, make sure you understand what each part of the equation means.

STEP 2

1. Find the time of maximum height.2. Calculate the maximum height.3. Determine the platform height.

STEP 3

Alright, so we've got this awesome equation h(t)=−5(t−0.3)2+110h(t) = -5(t - 0.3)^2 + 110h(t)=−5(t−0.3)2+110 where ttt is the time and h(t)h(t)h(t) is how high the jumper is at that time.We want to find the *biggest* value of h(t)h(t)h(t), the **maximum height**.

STEP 4

Notice that pesky negative sign in front of the 5?That tells us the squared term will *always* be negative (or zero) since anything squared is positive (or zero).Multiplying by −5-5−5 makes the whole term −5(t−0.3)2-5(t - 0.3)^2−5(t−0.3)2 negative (or zero).

STEP 5

Since we're *subtracting* that term from 110, to get the *biggest* possible value for h(t)h(t)h(t), we want to subtract the *smallest* possible amount.The smallest −5(t−0.3)2-5(t - 0.3)^2−5(t−0.3)2 can be is **zero**, which happens when t−0.3=0t - 0.3 = 0t−0.3=0.

STEP 6

STEP 7

Now, let's plug our **time of maximum height**, t=0.3t = 0.3t=0.3, back into our equation to find out just how high that is!

STEP 8

STEP 9

The platform height is just the jumper's height at the very beginning, when t=0t = 0t=0.Let's plug that into our equation!

STEP 10

STEP 11

a) The bungee jumper reaches maximum height at 0.3\mathbf{0.3}0.3 seconds. b) The maximum height reached is 110\mathbf{110}110 meters. c) The platform height is 109.55\mathbf{109.55}109.55 meters.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We're finding out when a bungee jumper reaches their highest point, how high that is, and how high the platform is! Watch out! Don't mix up time and height!
Also, make sure you understand what each part of the equation means.

1. Find the time of maximum height.
2. Calculate the maximum height.
3. Determine the platform height.

Alright, so we've got this awesome equation $h(t) = -5(t - 0.3)^2 + 110$ where $t$ is the time and $h(t)$ is how high the jumper is at that time.
We want to find the biggest value of $h(t)$ , the maximum height.

Notice that pesky negative sign in front of the 5?
That tells us the squared term will always be negative (or zero) since anything squared is positive (or zero).
Multiplying by $-5$ makes the whole term $-5(t - 0.3)^2$ negative (or zero).

Since we're subtracting that term from 110, to get the biggest possible value for $h(t)$ , we want to subtract the smallest possible amount.
The smallest $-5(t - 0.3)^2$ can be is zero, which happens when $t - 0.3 = 0$ .

Now, let's plug our time of maximum height, $t = 0.3$ , back into our equation to find out just how high that is!

The platform height is just the jumper's height at the very beginning, when $t = 0$ .
Let's plug that into our equation!

a) The bungee jumper reaches maximum height at $\mathbf{0.3}$ seconds. b) The maximum height reached is $\mathbf{110}$ meters. c) The platform height is $\mathbf{109.55}$ meters.