Question

A carnival uses two baskets hanging from springs at different heights. Next to the higher basket is a pile of baseballs. Next to the lower basket is a pile of golf balls . The object of the game is to add the same number of balls to each basket so that the baskets have the same height. But there is a catch - you only get one chance. What is the secret to winning the game ? You carefully observe the game and record the height of each basket in relation to the number of balls in it. Your observations are recorded in the tables to the right.

Answer 1

*I've tried looking up to see if I can find what the part B and c of this question is, but unfortunately, I can't find them. However, I have tried answering this question by answering part a, and also going ahead to answer how to state and explain the secrets of winning the game. I'm pretty sure most of the questions would be answered in the process.

Answer/Step-by-sep explanation:

Using an equation, in slope-intercept form, we can derive an equation that models the relationship of the height of each basket and the number of balls it contains.

The slope-intercept form is given as: $y = mx + b$, where,

m = slope/rate of change

b = y-intercept/strating value/height of the basket when it's empty.

✍️Baseball Equation:

Using two pairs, (0, 54) and (5, 39) from the given table of values,

Slope/rate of change (m) = $\frac{y_2 - y_1}{x_2 - x_1} = \frac{39 - 54}{5 - 0} = \frac{-15}{5} = -3$.

y-intercept, b, = the starting value, or the value of y when x = 0. Therefore, b = 54

This means that the baseball basket was at a height of 54 units when it was empty.

m = -3, means the baseball basket kept reducing at an average rate of -3 units in height as each ball was added.

To derive the baseball equation, substitute m = -3, and b = 54 into $y = mx + b$.

✅Thus, base ball equation would be:

$y = -3x + 54$

✍️Golf Ball Equation:

Using two pairs, (0, 45) and (5, 35) from the given table of values,

Slope/rate of change (m) = $\frac{y_2 - y_1}{x_2 - x_1} = \frac{35 - 45}{5 - 0} = \frac{-10}{5} = -2$.

y-intercept, b, = the starting value, or the value of y when x = 0. Therefore, b = 45

This means that the golf ball basket was at a height of 45 units when it was empty.

m = -2, means the golf ball basket kept reducing at an average rate of -2 units in height as each ball was added.

To derive the baseball equation, substitute m = -2, and b = 45 into $y = mx + b$.

✅Thus, golf ball equation would be:

$y = -2x + 45$

Now, to win the game, we have to find out how many number of exact balls (x) we need to add in each basket equally, for both baskets to be at the same height.

To do this, set the equation of the baseball equal to that of the golf ball.

Thus:

$-3x + 54 = -2x + 45$

Collect like terms

$-3x + 2x = -54 + 45$

$-x = -9$

Divide both sides by -1

$x = 9$

✅To win the game, add 9 balls each to both basket to make the height of both baskets equal.

Let's check to see if both baskets will yield the same height if we add 9 balls each basket.

✍️Height (y) of Golf ball basket if we add 9 balls (x):

Substitute x = 9 into $y = -2x + 45$

$y = -2(9) + 45 = -18 + 45$

$y = 27 units$

✍️Height (y) of Baseball basket if we add 9 balls (x):

Substitute x = 9 into $y = -3x + 54$

$y = -3(9) + 54 = -27 + 54$

$y = 27 units$

✅As we can see, both baskets will be at the same height of 27 units when we add 9 balls to each basket.

The game will be won if we do this.