Question

1. create an algebraic expression to represent the area/volume for the following shapes using the dimensions given. a) a = lw b) a = (bh)/2 c) v = lwh (with images of a rectangle with dimensions 12x²y² and 5x⁷y⁵, a triangle with base 4y³ and height 10y⁷, and a rectangular prism with length 4x²y³, width 3x⁴y⁵, and height 8x³)

Explanation:

Part (a)

Step1: Identify formula and values

The formula for the area of a rectangle is \( A = lw \), where \( l = 12x^{2}y^{2} \) and \( w = 5x^{7}y^{5} \).

Step2: Multiply the length and width

Multiply the coefficients and add the exponents of like variables: \( A=(12\times5)x^{2 + 7}y^{2+5} \)

Step3: Simplify the expression

Calculate the product of the coefficients and simplify the exponents: \( A = 60x^{9}y^{7} \)

Part (b)

Step1: Identify formula and values

The formula for the area of a triangle is \( A=\frac{bh}{2} \), where \( b = 4y^{3} \) and \( h = 10y^{7} \).

Step2: Multiply the base and height

Multiply the coefficients and add the exponents of like variables: \( bh=(4\times10)y^{3 + 7}=40y^{10} \)

Step3: Divide by 2

Divide the product of \( bh \) by 2: \( A=\frac{40y^{10}}{2} \)

Step4: Simplify the expression

Calculate the division: \( A = 20y^{10} \)

Part (c)

Step1: Identify formula and values

The formula for the volume of a rectangular prism is \( V = lwh \), where \( l = 4x^{2}y^{3} \), \( w = 3x^{4}y^{5} \), and \( h = 8x^{3} \).

Step2: Multiply the length, width, and height

Multiply the coefficients and add the exponents of like variables: \( V=(4\times3\times8)x^{2+4 + 3}y^{3+5} \)

Step3: Simplify the expression

Calculate the product of the coefficients and simplify the exponents: \( V = 96x^{9}y^{8} \)

Answer:

s:
a) \( \boldsymbol{60x^{9}y^{7}} \)
b) \( \boldsymbol{20y^{10}} \)
c) \( \boldsymbol{96x^{9}y^{8}} \)