If the quarterback in the following cartoon is right, what is the greatest number of people that

step 1 Okay, here's my football field. Here's the quarterback. And the quarterback throws the ball to t

If the quarterback in the following cartoon is right, what...

Key Concepts

Constraint Analysis

Constraint analysis focuses on identifying and understanding all limits imposed by a problem's conditions. Interpreting the statement 'if the quarterback is right' requires considering how this assumption restricts or informs the count, which is central to solving maximum or minimum value problems in combinatorics and logic.

Counting and Combinatorial Principles

These are the fundamental ideas used to enumerate possibilities under prescribed conditions. When a problem asks for the maximum number of people or items, one applies combinatorial strategies to ensure all possible cases are considered while adhering to any stipulated constraints.

Logical Deduction

This involves drawing conclusions from a given set of statements or conditions. In puzzles, the solver uses the information provided—explicit or implicit—to determine how various constraints interact and what conclusions can be obtained. This reasoning is crucial when a condition such as 'if X is right' affects the possible outcomes.

Transcript

00:02 Okay, here's my football field.

00:06 Here's the quarterback.

00:08 And the quarterback throws the ball to the receiver.

00:13 And the question is just how far did the ball go? or how far apart are these two players? so we know that the quarterback is on the 10 -yard line and they're 20 yards from the sideline.

00:28 So let me label that sideline distance too.

00:34 And then we know that the receiver is 5 yards from the right sideline.

00:42 So let me move that label out of the way.

00:50 So let's make some coordinates.

00:54 So how about we call the origin right here? that'll be zero zero.

01:02 So then what would the quarterbacks coordinate be? well, their x value is 20.

01:13 They're 20 to the right of my origin and they're 10 yards above the origin.

01:20 So the coordinate is 20, 10.

01:22 Now, how about the receiver? well, we were given this distance from the right sideline, but we really need to know how far they are from the left.

01:34 And we're given that the whole field is 53 and one -third yards across.

01:45 So to get the distance from the left, we've got to do 53 and a third minus 5, which what would that be? well, 53 minus 5 is 48, so it's going to be 48 okay, so their coordinate would be 48 and a third, and then they were on the 45 yard line, so they're 45 units above the origin.

02:22 Okay, so now that we have two coordinates, we can find the distance.

02:28 So the distance formula is the change in y coordinates squared plus the change in x coordinate coordinate squared, and then we take the square root of the sum of those two squares.

03:00 Okay, so let's plug in what we know.

03:06 So the y2, let's use, here i'll just label them, let's call the receiver's coordinate x2, y2, and the quarterback will be x1, y1.

03:21 So y2 is going to be 45, y1 is going to be 10, x2 is going to be 48 and a third, and x1 is going to be 20.

03:38 Okay, so let's do that subtraction.

03:41 So we'll have 35 squared plus 28 and a third squared.

03:55 Okay, so let's do, i'll grab my calculator, so i'll do 35 squared plus 28 and a third squared.

04:12 I don't want to mess that up, so i'll do that separately.

04:16 So i get 1225 for this...