Answer:
Step-by-step explanation:
Given that :
the side of the square = 90ft
The speed of the runner = 31 ft/sec
By the time the runner is halfway to the first base; the distance covered by the runner in time(t) is (31 t) ft and the distance half the base = 90/2 = 45 ft
Thus; 31 t = 45
t = 45/31
From the second base ; the distance is given as:
P² = (90)² + (90 - 31t )²
P = [tex]\sqrt{(90)^2 + (90 - 31t )^2}[/tex]
By differentiation with time;
[tex]\dfrac{dP}{dt} =\dfrac{1}{ 2 \sqrt{90^2 +(90-31t)^2} } *(0+ 2 (90-31t)(0-31))[/tex]
[tex]\dfrac{dP}{dt} =\dfrac{1}{ 2 \sqrt{90^2 +(90-31t)^2} } * 2 (-31)(90-31t)[/tex]
At t = 45/31
[tex]\dfrac{dP}{dt} =\dfrac{1}{ 2 \sqrt{90^2 +45^2} } * 2 (-31)(45)[/tex]
[tex]\dfrac{dP}{dt} =\dfrac{-35*45}{100.623}[/tex]
= - 13.86 ft/sec
Hence, we can conclude that as soon as the runner is halfway to the first base, the distance to the second base is therefore decreasing by 13.86 ft/sec
b) The distance from third base can be expressed by the relation:
q² = (31t)² + (90)²
[tex]q = \sqrt{(31t)^2+(90)^2}[/tex]
By differentiation with respect to time:
[tex]\dfrac{dq}{dt} = \dfrac{1}{2\sqrt{90^2 + (31)t^2} } *(0+31^2 + 2t)[/tex]
At t = 45/31
[tex]\dfrac{dq}{dt} = \dfrac{1}{2\sqrt{90^2 + 45^2} } *(0+31^2 + \frac{45}{31})[/tex]
[tex]= \dfrac{31*45}{100.623}[/tex]
[tex]= 13.86 \ ft/sec[/tex]
Thus, the rate at which the runner's distance is from the third base is increasing at the same moment of 13.86 ft/sec. So therefore; he is moving away from the third base at the same speed to the first base)
a) The distance from second base is decreasing when the batter is halfway to first base at a rate of 13.9 feet per second.
b) The distance from third base is increasing when the batter is halfway to first base at a rate of 13.9 feet per second.
a) As the batter runs towards the first base, both the distance from second base and the length of the line segment PQ decrease in time. The distance from the second base is determined by Pythagorean theorem:
[tex]QS^{2} = QP^{2}+PS^{2}[/tex] (1)
By differential calculus we derive an expression for the rate of change of the distance from second base ([tex]\dot QS[/tex]), in feet per second:
[tex]2\cdot QS \cdot \dot{QS} = 2\cdot QP\cdot \dot{QP} + 2\cdot PS\cdot \dot {PS}[/tex]
[tex]\dot{QS} = \frac{QP\cdot \dot QP + PS\cdot \dot{PS}}{QS}[/tex]
[tex]\dot {QS} = \frac{QP\cdot \dot {QP}+PS\cdot \dot {PS}}{\sqrt{QP^{2}+PS^{2}}}[/tex] (2)
If we know that [tex]QP = 0.5L[/tex], [tex]PS = L[/tex], [tex]L = 90\,ft[/tex], [tex]\dot {QP} = -31\,\frac{ft}{s}[/tex] and [tex]\dot {PS} = 0\,\frac{ft}{s}[/tex], then the rate of change of the distance from second base is:
[tex]\dot {QS} = \frac{(45\,ft)\cdot \left(-31\,\frac{ft}{s} \right)}{\sqrt{(45\,ft)^{2}+(90\,ft)^{2}}}[/tex]
[tex]\dot {QS} \approx -13.864\,\frac{ft}{s}[/tex]
The distance from second base is decreasing when the batter is halfway to first base at a rate of 13.9 feet per second.
b) As the batter runs towards the first base, both the distance from third base increases and the distance from home increase in time. The distance from the third base is determined by Pythagorean theorem:
[tex]QT^{2} = HT^{2}+QH^{2}[/tex] (3)
By differential calculus we derive an expression for the rate of change of the distance from third base ([tex]\dot QT[/tex]), in feet per second:
[tex]2\cdot QT\cdot \dot{QT} = 2\cdot HT\cdot \dot {HT} + 2\cdot QH\cdot \dot {QH}[/tex]
[tex]\dot {QT} = \frac{HT\cdot \dot {HT}+QH\cdot \dot {QH}}{QT}[/tex]
[tex]\dot {QT} = \frac{HT\cdot \dot {HT}+QH\cdot \dot {QH}}{\sqrt{HT^{2}+QH^{2}}}[/tex]
If we know that [tex]HT = 90\,ft[/tex], [tex]QH = 45\,ft[/tex], [tex]L = 90\,ft[/tex], [tex]\dot{HT} = 0\,\frac{ft}{s}[/tex] and [tex]\dot {QH} = 31\,\frac{ft}{s}[/tex], then the rate of change of the distance from third base is:
[tex]\dot{QT} = \frac{(45\,ft)\cdot \left(31\,\frac{ft}{s} \right)}{\sqrt{(90\,ft)^{2}+(45\,ft)^{2}}}[/tex]
[tex]\dot{QT} \approx 13.864\,\frac{ft}{s}[/tex]
The distance from third base is increasing when the batter is halfway to first base at a rate of 13.9 feet per second.
We kindly invite to check this question on rates of change: https://brainly.com/question/13103052
