Answer:
c: car b was 10 miles east of car a
Step-by-step explanation:
Answer:
a line segment is an object with two endpoints , a ray is a part of a line that has no fixed starting point
Step-by-step explanation:
Answer:
Step-by-step explanation:
1. Given the integral function
, using trigonometric substitution, the substitution that will be most helpful in this case is substituting x as
i.e
.
All integrals in the form
are always evaluated using the substitute given where 'a' is any constant.
From the given integral,
where a = 7 in this case.
The substitute will therefore be ![x = 7 sin\theta](https://tex.z-dn.net/?f=x%20%3D%207%20sin%5Ctheta)
2.) Given ![x = 7 sin\theta](https://tex.z-dn.net/?f=x%20%3D%207%20sin%5Ctheta)
![\frac{dx}{d \theta} = 7cos \theta](https://tex.z-dn.net/?f=%5Cfrac%7Bdx%7D%7Bd%20%5Ctheta%7D%20%3D%207cos%20%5Ctheta)
cross multiplying
![dx = 7cos\theta d\theta](https://tex.z-dn.net/?f=dx%20%3D%207cos%5Ctheta%20d%5Ctheta)
3.) Rewriting the given integral using the substiution will result into;
![\int\limits {7\sqrt{49-x^{2} } } \, dx \\= \int\limits {7\sqrt{7^{2} -x^{2} } } \, dx\\= \int\limits {7\sqrt{7^{2} -(7sin\theta)^{2} } } \, dx\\= \int\limits {7\sqrt{7^{2} -49sin^{2}\theta } } \, dx\\= \int\limits {7\sqrt{49(1-sin^{2}\theta)} } } \, dx\\= \int\limits {7\sqrt{49(cos^{2}\theta)} } } \, dx\\since\ dx = 7cos\theta d\theta\\= \int\limits {7\sqrt{49(cos^{2}\theta)} } } \, 7cos\theta d\theta\\= \int\limits {7\{7(cos\theta)} }}} \, 7cos\theta d\theta\\](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B7%5Csqrt%7B49-x%5E%7B2%7D%20%7D%20%7D%20%5C%2C%20dx%20%5C%5C%3D%20%5Cint%5Climits%20%7B7%5Csqrt%7B7%5E%7B2%7D%20-x%5E%7B2%7D%20%7D%20%7D%20%5C%2C%20dx%5C%5C%3D%20%5Cint%5Climits%20%7B7%5Csqrt%7B7%5E%7B2%7D%20-%287sin%5Ctheta%29%5E%7B2%7D%20%7D%20%7D%20%5C%2C%20dx%5C%5C%3D%20%5Cint%5Climits%20%7B7%5Csqrt%7B7%5E%7B2%7D%20-49sin%5E%7B2%7D%5Ctheta%20%20%7D%20%7D%20%5C%2C%20dx%5C%5C%3D%20%5Cint%5Climits%20%7B7%5Csqrt%7B49%281-sin%5E%7B2%7D%5Ctheta%29%7D%20%20%20%7D%20%7D%20%5C%2C%20dx%5C%5C%3D%20%5Cint%5Climits%20%7B7%5Csqrt%7B49%28cos%5E%7B2%7D%5Ctheta%29%7D%20%20%20%7D%20%7D%20%5C%2C%20dx%5C%5Csince%5C%20dx%20%3D%207cos%5Ctheta%20d%5Ctheta%5C%5C%3D%20%5Cint%5Climits%20%7B7%5Csqrt%7B49%28cos%5E%7B2%7D%5Ctheta%29%7D%20%20%20%7D%20%7D%20%5C%2C%207cos%5Ctheta%20d%5Ctheta%5C%5C%3D%20%5Cint%5Climits%20%7B7%5C%7B7%28cos%5Ctheta%29%7D%20%20%20%7D%7D%7D%20%5C%2C%207cos%5Ctheta%20d%5Ctheta%5C%5C)
![= \int\limits343 cos^{2} \theta \, d\theta](https://tex.z-dn.net/?f=%3D%20%5Cint%5Climits343%20cos%5E%7B2%7D%20%20%5Ctheta%20%5C%2C%20d%5Ctheta)
Pennies Toby has = 178
Pennies Bella has = 190
Hence, Bella has more pennies by 190-178=12
Bella has 12 more pennies than Toby.