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kykrilka [37]
3 years ago
11

24 coloring books,60 crayons, and 84 markers can be packaged into at most how many Identical packages? How many will each packag

e contain?
Mathematics
1 answer:
bearhunter [10]3 years ago
4 0

Answer:

there can be 12 identical packages at most. each package will contain:

- 2 coloring books

- 5 crayons

- 7 markers

Step-by-step explanation:

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In evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows:
BabaBlast [244]

Answer

a=0, b=2

g_1(x)=\frac{5x}{2},  g_2(x)=7-x

Step-by-step explanation:

Given that

\int \int   Df(x,y)dA=\int_0 ^5\int _0 ^ {\frac {2y}{5}} f(x,y)dxdy+\int_5^7\int_0^{7-y} f(x,y)dxdy\; \cdots (i)

For the term  \int_0 ^5\int _0 ^ {\frac {2y}{5}} f(x,y)dxdy.

Limits for x is from x=0 to x=\frac {2y}{5} and for y is from y=0 to y=5  and the region D, for this double integration is the shaded region as shown in graph 1.

Now, reverse the order of integration, first integrate with respect to y then with respect to x . So, the limits of y become from y=\frac{5x}{2} to y=5 and limits of x become from x=0 to x=2 as shown in graph 2.

So, on reversing the order of integration, this double integration can be written as

\int_0 ^5\int _0 ^ {\frac {2y}{5}} f(x,y)dxdy=\int_0 ^2\int _ {\frac {5x}{2}}^5 f(x,y)dydx\; \cdots (ii)

Similarly, for the other term  \int_5 ^7\int _0 ^ {7-y} f(x,y)dxdy.

Limits for x is from x=0 to x=7-y and limits for y is from y=5 to y=7  and the region D, for this double integration is the shaded region as shown in graph 3.

Now, reverse the order of integration, first integrate with respect to y then with respect to x . So, the limits of y become from y=5 to y=7-x and limits of x become from x=0 to x=2 as shown in graph 4.

So, on reversing the order of integration, this double integration can be written as

\int_5 ^7\int _0 ^ {7-y} f(x,y)dxdy=\int_0 ^2\int _5 ^ {7-x} f(x,y)dydx\;\cdots (iii)

Hence, from equations (i), (ii) and (iii) , on reversing the order of integration, the required expression is

\int \int   Df(x,y)dA=\int_0 ^2\int _ {\frac {5x}{2}}^5 f(x,y)dydx+\int_0 ^2\int _5 ^ {7-x} f(x,y)dydx

\Rightarrow \int \int   Df(x,y)dA=\int_0 ^2\left(\int _ {\frac {5x}{2}}^5 f(x,y)+\int _5 ^ {7-x} f(x,y)\right)dydx

\Rightarrow \int \int   Df(x,y)dA=\int_0 ^2\int _ {\frac {5x}{2}}^{7-x} f(x,y)dydx\; \cdots (iv)

Now, compare the RHS of the equation (iv) with

\int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y)dydx

We have,

a=0, b=2, g_1(x)=\frac{5x}{2} and g_2(x)=7-x.

3 0
3 years ago
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