Questions: A ramp in a skate park can be modelled by a piecewise linear function. Each section of the ramp is a straight line, and the coordinates (measured in metres) of the endpoints of the sections are shown. The middle section of the ramp is horizontal. The formula for this piecewise linear function is given by f(x) = mx for 0 ≤ x ≤ k l for k ≤ x ≤ 5 px+q for 5 ≤ x ≤ r Find the values of: 1. m 2. l 3. k 4. a 5. b 6. p 7. q 8. r.

A ramp in a skate park can be modelled by a piecewise linear function. Each section of the ramp is a straight line, and the coordinates (measured in metres) of the endpoints of the sections are shown.

The middle section of the ramp is horizontal.
The formula for this piecewise linear function is given by

f(x) =
mx for 0 ≤ x ≤ k
l for k ≤ x ≤ 5
px+q for 5 ≤ x ≤ r

Find the values of:
1. m
2. l
3. k
4. a
5. b
6. p
7. q
8. r.

Transcript text: A ramp in a skate park can be modelled by a piecewise linear function. Each section of the ramp is a straight line, and the coordinates (measured in metres) of the endpoints of the sections are shown. The middle section of the ramp is horizontal. The formula for this piecewise linear function is given by \[ f(x)=\left\{\begin{array}{ll} m x & 0 \leq x \leq k \\ l & k \leq x \leq 5 \\ p x+q & 5 \leq x \leq r \end{array}\right. \] Find the values of: 1 m ii $l$ iii $k$ iv $a$ $v \quad b$ vi $p$ vii $q$ viii $r$.

Solution

Solution Steps

Step 1: Find _m_

The first section of the function goes through the points (0,0) and (3,0.6) and is defined by _f(x) = mx_. Substituting the point (3,0.6):

0.6 = _m_ * 3 _m_ = 0.6/3 = 0.2

Step 2: Find _l_ and _k_

The middle section of the ramp is horizontal and thus _f(x) = l_. The middle section spans from _x=k_ to _x=5_. Since the point (3, 0.6) is the end of the first section and the beginning of the horizontal section, _k=3_. Since it's horizontal, _l=0.6_.

Step 3: Find _a_ and _b_

The point (_a_,_b_) is at the intersection of the middle section and the third section. Because the middle section ends at x=5, we know _a_=5. Also, because the middle section is _f(x)=0.6_, we know _b_=0.6. Therefore, (_a_,_b_) is (5,0.6).