Question

1. yoshi is training for a hot dog eating contest at the kappa tau annual carnival. the chart below display how many hot dogs he consumes during his training in a certain amount of time.

hot dogs | 3 | 6 | 9 | 12
time | 4 | 6 | 8 | 10
which equation best represents the data given? (hint: which are the independent/dependent variables?)
a. t = 2/3h c. t = 2/3h + 2
b. h = 2/3t + 2 d. h = 2/3t
how many hot dogs would yoshi consume in 15 minutes?
e. 8 g. 19.5
f. 12.4 h. 21

Explanation:

First Sub - Question: Which equation best represents the data given?

Step 1: Identify Independent and Dependent Variables

Time (T) is the independent variable (it's the input, we can control or measure it), and Hot Dogs (H) is the dependent variable (it depends on the time). So we need an equation in the form \(H = mT + b\) (slope - intercept form of a linear equation, where \(m\) is the slope and \(b\) is the y - intercept).

Step 2: Calculate the Slope (\(m\))

We use the formula for slope \(m=\frac{H_2 - H_1}{T_2 - T_1}\). Let's take two points from the table. For example, when \(T_1 = 4\), \(H_1=3\) and when \(T_2 = 6\), \(H_2 = 6\). Then \(m=\frac{6 - 3}{6 - 4}=\frac{3}{2}\)? Wait, no, wait. Wait, maybe I mixed up H and T. Wait, the table is Hot Dogs (H): 3, 6, 9, 12 and Time (T): 4, 6, 8, 10. Let's take two points \((T_1,H_1)=(4,3)\) and \((T_2,H_2)=(6,6)\). Then the slope \(m=\frac{H_2 - H_1}{T_2 - T_1}=\frac{6 - 3}{6 - 4}=\frac{3}{2}\)? No, that's not right. Wait, maybe the equation is in terms of T as a function of H? Wait, no, let's check the options. The options are:
a. \(T=\frac{2}{3}H\)
b. \(H=\frac{2}{3}T + 2\)
c. \(T=\frac{2}{3}H+2\)
d. \(H=\frac{2}{3}T\)

Let's test option b: \(H=\frac{2}{3}T + 2\). When \(T = 4\), \(H=\frac{2}{3}(4)+2=\frac{8}{3}+2=\frac{8 + 6}{3}=\frac{14}{3}\approx4.67
eq3\). Not good.

Test option d: \(H=\frac{2}{3}T\). When \(T = 4\), \(H=\frac{2}{3}(4)=\frac{8}{3}\approx2.67
eq3\).

Test option b again? Wait, no, let's take \(T = 4\), \(H = 3\). Let's plug into option b: \(3=\frac{2}{3}(4)+2=\frac{8}{3}+2=\frac{8 + 6}{3}=\frac{14}{3}\approx4.67\). No.

Wait, maybe we should consider T as a function of H. Let's take \(H = 3\), \(T = 4\) and \(H = 6\), \(T = 6\). The slope \(m=\frac{T_2 - T_1}{H_2 - H_1}=\frac{6 - 4}{6 - 3}=\frac{2}{3}\). Now, using the point - slope form \(T - T_1=m(H - H_1)\). Using \((H_1 = 3,T_1 = 4)\), we get \(T-4=\frac{2}{3}(H - 3)\). Then \(T-4=\frac{2}{3}H-2\), so \(T=\frac{2}{3}H + 2\). Let's test this equation. When \(H = 3\), \(T=\frac{2}{3}(3)+2=2 + 2=4\) (correct). When \(H = 6\), \(T=\frac{2}{3}(6)+2=4 + 2=6\) (correct). When \(H = 9\), \(T=\frac{2}{3}(9)+2=6 + 2=8\) (correct). When \(H = 12\), \(T=\frac{2}{3}(12)+2=8 + 2=10\) (correct). So the correct equation is \(T=\frac{2}{3}H + 2\)? Wait, no, the question is about which equation best represents the data. Wait, the options: option c is \(T=\frac{2}{3}H+2\), which we just verified. Wait, but let's check the first part again. Wait, maybe I made a mistake in variable assignment. The problem says "Which equation best represents the data given? (HINT: Which are the Independent/dependent variables?)". Time is independent (T), Hot Dogs is dependent (H). But when we derived \(T=\frac{2}{3}H + 2\), that's T as a function of H. But let's check the options again.

Wait, let's check option b: \(H=\frac{2}{3}T+2\). When \(T = 4\), \(H=\frac{8}{3}+2=\frac{14}{3}\approx4.67\) (wrong). Option d: \(H=\frac{2}{3}T\), when \(T = 4\), \(H=\frac{8}{3}\approx2.67\) (wrong). Option a: \(T=\frac{2}{3}H\), when \(H = 3\), \(T = 2\) (wrong, since T should be 4). Option c: \(T=\frac{2}{3}H+2\), when \(H = 3\), \(T=\frac{2}{3}(3)+2=2 + 2=4\) (correct). When \(H = 6\), \(T=\frac{2}{3}(6)+2=4 + 2=6\) (correct). When \(H = 9\), \(T=\frac{2}{3}(9)+2=6 + 2=8\) (correct). When \(H = 12\), \(T=\frac{2}{3}(12)+2=8 + 2=10\) (correct). So the correct equation is option c: \(T=\frac{2}{3}H + 2\)? Wait, no, the question is about the equation that represents the data. Wait, maybe I had the variables reversed. Wait, the table is Hot Dogs (H) and Time (T). So if we want H as a function of T, let's solve \(T=\frac{2…

Step 1: Use the correct equation

We found that the equation is \(T=\frac{2}{3}H + 2\) (where T is time in minutes and H is the number of hot dogs). We need to find H when T = 15.

Step 2: Solve for H

Start with \(T=\frac{2}{3}H + 2\). Substitute T = 15:
\(15=\frac{2}{3}H+2\)
Subtract 2 from both sides: \(15 - 2=\frac{2}{3}H\)
\(13=\frac{2}{3}H\)
Multiply both sides by \(\frac{3}{2}\): \(H = 13\times\frac{3}{2}=\frac{39}{2}=19.5\)

Answer:

First Sub - Question: c. \(T=\frac{2}{3}H + 2\)
Second Sub - Question: g. 19.5