Joe's heart would beat 66 times in 60 seconds
Problem-Solving
To solve this problem, we can use different approaches: a proportion, a linear function, and a conversion. Let's go through each approach step-by-step.
(a) Proportion:
A proportion is a statement that two ratios are equal. In this case, we can set up a proportion to find the number of heartbeats in 60 seconds.
Let's set up the proportion:
11 beats / 10 seconds = x beats / 60 seconds
To solve this proportion, we can cross-multiply:
11 * 60 = 10 * x
660 = 10x
Now, divide both sides by 10:
660 / 10 = x
x = 66
So, at the same rate, Joe's heart would beat 66 times in 60 seconds.
(b) Linear Function:
A linear function can be used to find the relationship between two variables. In this case, we can use a linear function to find the number of heartbeats in 60 seconds based on the given information.
Let's set up the linear function:
y = mx + b
In this case, y represents the number of heartbeats, and x represents the number of seconds. We are given that the heart beats 11 times in 10 seconds. So, we can find the slope (m) as the rate of change:
m = (11 - 0) / (10 - 0) = 11/10
Now, let's find the y-intercept (b) by plugging in the values from the given information:
11 = (11/10) * 10 + b
11 = 11 + b
b = 0
The linear function is now:
y = (11/10) * x
To find the number of heartbeats in 60 seconds, we can substitute x = 60 into the function:
y = (11/10) * 60
y = 66
So, using the linear function, Joe's heart would beat 66 times in 60 seconds.
(c) Conversion:
The conversion method involves converting the given rate to the desired rate.
We are given that Joe's heart beats 11 times in 10 seconds. To find the number of heartbeats in 60 seconds, we can set up a conversion using ratios.
Let's set up the conversion:
11 beats / 10 seconds = x beats / 60 seconds
To find x, we can cross-multiply:
11 * 60 = 10 * x
660 = 10x
Now, divide both sides by 10:
660 / 10 = x
x = 66
So, using the conversion method, Joe's heart would beat 66 times in 60 seconds.
(d) Explanation:
All three methods (proportion, linear function, conversion) give the same answer because they are all mathematically equivalent and based on the same given information.
The proportion method uses ratios to find the unknown value by setting up an equal proportion.
The linear function method uses a linear equation to find the relationship between the variables and calculates the desired value based on the given information.
The conversion method uses ratios to convert the given rate to the desired rate.
In all cases, the final result is 66, which means Joe's heart would beat 66 times in 60 seconds.