Question

1. model rockets are created in various sizes. the height of a rocket in inches, h(x), depends on the radius of the base of the rocket in inches, x. use the table to write an equation for h(x) that outputs the height of the rocket with a base radius of x.

problems 8–9: the function w(t) models the weight of a pumpkin, in pounds, as a function of how many months, t, it has been growing. explain the meaning of each statement.

1. w(2) = 5
2. w(6) > w(4)

spiral review
problems 10–11: here are the first four figures in a pattern.

1. how many dots will be in figure 5?
2. write an equation for the number of dots, d(n), in figure n.

reflection

1. circle a problem you want to talk to a classmate about.
2. use this space to ask a question or share something you’re proud of

Explanation:

Problem 7

Step 1: Check if it's linear

Calculate the slope between two points. Let's take \((x_1, h(x_1))=(1, 5)\) and \((x_2, h(x_2))=(3, 13)\). The slope \(m=\frac{13 - 5}{3 - 1}=\frac{8}{2}=4\).

Step 2: Find the y - intercept

Use the point - slope form \(y - y_1=m(x - x_1)\). Using the point \((1,5)\) and \(m = 4\), we have \(h(x)-5 = 4(x - 1)\).
Simplify: \(h(x)-5=4x-4\), so \(h(x)=4x + 1\).
We can check with another point. For \(x = 5\), \(h(5)=4\times5+1=21\), which matches the table. For \(x = 10\), \(h(10)=4\times10 + 1=41\), which also matches.

The function \(w(t)\) gives the weight of a pumpkin (in pounds) as a function of the number of months \(t\) it has been growing. The statement \(w(2)=5\) means that when the pumpkin has been growing for \(t = 2\) months, its weight is \(5\) pounds.

The function \(w(t)\) represents the weight of the pumpkin (in pounds) based on the number of months \(t\) of growth. The statement \(w(6)>w(4)\) means that the weight of the pumpkin when it has been growing for 6 months is greater than the weight of the pumpkin when it has been growing for 4 months.

Answer:

\(h(x)=4x + 1\)

Problem 8