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Skateboard Ramp

GMAT-11WLJQ

The figure below models a skateboard ramp made of two smoothly connecting curves. At position 0 feet, the height is 0, and the slope of the ramp is 0. At position 100 feet, the height is 20 feet. The two curves constructing the ramp meet at this location. The ramp increases in height for all values of $x$, where $x$ is the horizontal position of any point on the ramp.

If $R(x)$ denotes the height of the ramp at position $x$, then ${ R }_{ 1 }^{ }(x)=a{ x }^{ 2 }$ is an equation that describes the ramp height for positions between $x = 0$ feet and $x = 100$ feet, and the equation ${ R }_{ 2 }^{ }(x)=b\sqrt { x } +c$ describes the ramp height for positions between $x = 100$ feet and $x = 144$ feet.

Note that $a$, $b$, and $c$ are numerical values not yet established.

The function ${ S }_{ 1 }^{ }(x)=2ax$ measures the slope of the ramp for positions between $x = 0$ and $x = 100$, and the function ${ S }_{ 2 }^{ }(x)=\cfrac { b }{ 2\sqrt { x } } $ measures the slope of the ramp for positions between $x = 100$ and $x = 144$.

Find the value of $c$ for this ramp.

Based on the height and slope of the ramp at $x = 144$​ feet, the ramp will be extended by a third section that will be connected smoothly, following the same slope along a linear path. The path must extend far enough to reach a height of 39.2 feet, which will be the end of the ramp.

What will be the horizontal position of the right endpoint of this third segment?

Value of c

Horizontal endpoint

Value of c

Horizontal endpoint

0.002

Value of c

Horizontal endpoint

0.25

Value of c

Horizontal endpoint

8

Value of c

Horizontal endpoint

-60

Value of c

Horizontal endpoint

20

Value of c

Horizontal endpoint

36

Value of c

Horizontal endpoint

9.6

Value of c

Horizontal endpoint

153.6

Value of c

Horizontal endpoint

180