Problem of the Week Archive
Soccer Superstar – July 13, 2015
Problems & Solutions
John is the captain of his school’s soccer team and the highest scoring player. So far
this season, John’s team has played 10 soccer games. He scored 2, 1, 1, 3, 2, 0, 1, 4,
1, 2 in each of the 10 games, respectively. The school record for the average number of
goals scored per game in a single season is 1.91 goals. John wants to break this
school record, what is the minimum combined number of goals he must score in the
remaining three games of the season?
In order to average exactly 1.91 goals scored per game for the entire season, the total number of goals John scores divided by
13 must equal 1.91. In the 10 games played so far, John has scored a total of 2 + 1 + 1 + 3 + 2 + 0 + 1 + 4 + 1 + 2 = 17
goals. Let x represent the total number of additional goals needed to average exactly 1.91 goals scored per game. We can set
up the equation (17 + x)/13 = 1.91. Solving for x, we find that x = 1.91 × 13 −17 = 7.83 goals. So, by scoring a total of 7.83
goals over the next three games, John would tie the school record. However, John cannot score 0.83 goal, and he wants to
break the school record, not tie it. Therefore, he must score the minimum whole number of goals that is greater than 7.83 goals,
which is 8 goals.
John was fouled by the other team and is
lined up to take a penalty kick. The goalie
is positioned just inside the goal line and
centered with the goal, which measures
8 feet by 24 feet. In the time it takes for
John’s shot to reach the goal, the goalie
is able to defend the area, indicated here
by dotted lines, composed of a
rectangle, which is 12 feet wide and extends to a height of 2 feet above the ground, and a
semicircle whose diameter coincides with the top side of the rectangle, as shown. The undefended
area, at which John should aim his penalty kick, represents what percent of the total area of the
goal? Express your answer to the nearest whole number.
The total area of the goal is 8 ft × 24 ft = 192 ft2. The area the goalie defends has two parts, the rectangular area and the
semicircular area. The rectangular area covered is 12 ft × 2 ft = 24 ft2. The semicircular area is ½ × π × 62 = 18π ft2. The
portion of the total goal area that is undefended and where John should aim his kick is 1 – ((24 + 18π)/192) = 0.58. The
percent of the total area, therefore, is 0.58 × 100 = 58%.
John and a defender from the other team are running down the field towards the other team’s goal.
John’s teammate passes the ball and it lands 10 yards in front of the opponent, John is running
3 feet behind so he will not be called offside. If the defender is running at a rate of 20 feet per
second, what is the minimum speed John must run in order to get to the passed ball first? Express
your answer to the nearest whole number.
If the defender is running at 20 feet per second and the ball is 10 yards in front of him, then it will take the defender
10 yards × 3 ft/yard ÷ 20 ft/s = 1.5 seconds to reach the ball. If John wants to arrive at the ball at the exact same time the
defender does, he will have to run exactly (10 yards × 3 ft/yard + 3 ft) ÷ 1.5 sec = 22 feet per second. Since John wants to
beat his opponent to the ball, however, he will need to run at a minimum speed of 23 feet per second.